At least one person was killed and 17 injured by Israeli forces in southern Lebanon on Sunday, Lebanese officials said, as the 60-day deadline for both Hezbollah and Israel to withdraw from the south expired and thousands of Lebanese displaced by the war poured into roads leading south back to their homes.

The agreement, which was signed in November, stipulated that both Hezbollah and Israel withdraw their forces from southern Lebanon, while the Lebanese Army and U.N. peacekeepers would be deployed to secure the area. If the truce held, negotiators hoped the agreement would become permanent, returning a measure of calm to a turbulent region and bringing a close to Lebanon’s deadliest war in decades.

But as the deadline passed on Sunday, a very different scenario was taking shape.

Israeli forces remained in parts of southern Lebanon, stoking fears among Lebanese of a sustained Israeli occupation and renewed hostilities between Israel and Hezbollah. And Israeli officials were warning Lebanese not to return to their homes in many towns and villages in southern Lebanon.

“In the near future, we will continue to inform you about the places to which you can return,” Avichai Adraee, the Arabic spokesman of the Israeli military, posted on social media Sunday morning. “Until further notice, all previously published instructions remain in effect.”

Lebanon’s Health Ministry said that the 17 people injured on Sunday morning had been trying to enter their two villages, Kfar Kila and Houla, along the border when they encountered Israeli attacks. Residents of both towns had called for their neighbors to gather early Sunday morning and head to their homes in a convoy, despite the warnings from Israel.

The Israeli military did not immediately respond to a request for comment.

The situation poses a critical test for Lebanon’s new leaders, President Joseph Aoun and Prime Minister-designate Nawaf Salam, as they seek to wrestle back political control from Hezbollah, the country’s dominant political and military force.

Any prolonged Israeli occupation of southern Lebanon could breathe new life into Hezbollah, a group that was founded to liberate Lebanon from Israeli occupation and that has portrayed itself as the only force capable of protecting Lebanon’s borders, experts say.

It also threatens to derail the current political momentum in Lebanon, where for the first time in decades there is a serious push to consolidate all military power within the state, and do away with Hezbollah’s justification for its vast arsenal.

The focus in Lebanon now is toward “disarming Hezbollah and transitioning from the era in which Hezbollah was seen as having the right to acquire weapons,” said Mohanad Hage Ali, deputy director for research at the Carnegie Middle East Center in Beirut. Any prolonged Israeli occupation “would put the breaks on that momentum, which is happening organically,” he added.

Israeli officials have cited concerns that Hezbollah remains active in southern Lebanon and doubts about the Lebanese Army’s ability to stymie the group. Hezbollah officials did not respond to these accusations but said they were “committed” to upholding the terms of the truce.

On Saturday, Lebanese Army officials said they were prepared to complete their deployment in the south but had been delayed “as a result of the procrastination in the withdrawal by the Israeli enemy,” according to a Lebanese Army statement.

The 60-day truce took effect more than a year after Hezbollah began firing rockets toward Israeli positions in solidarity with its ally Hamas, the Palestinian militant group in Gaza that led the Oct. 7, 2023, attack on Israel. Israel retaliated by assassinating Hezbollah’s leadership, leveling towns and villages along the border and invading southern Lebanon.

Even before Sunday’s deadline, thousands of Lebanese who were displaced by the war from homes along the southern border were preparing to return home. On Saturday, the main highway leading from the capital, Beirut, to southern Lebanon was packed with cars even as people received automated phone calls from the Israeli military on Saturday warning them not to return home.

“You are forbidden to go back to your home until further notice,” the automated voice said. “Anyone driving south is putting their life at risk.”

Israeli forces appeared to be continuing efforts that persisted during the 60-day truce to bulldoze and block roads between some villages in southern Lebanon, according to local media. Israel currently occupies roughly 70 percent of the areas that it captured after invading Lebanon last fall, according to the United Nations peacekeeping force in southern Lebanon.

The Lebanese Army also warned about the dangers of unexploded ordnance in some villages and towns. Still, few Lebanese appeared dissuaded from returning home.

“The people of the land will force their way in,” said Abed al Karim Hasan, a banana farmer in Maaliye, a village in southern Lebanon, whose home was destroyed during the war. “If I had a house there, I would go there first thing tomorrow.”

Hezbollah has not said how it plans to respond to Israel’s continued occupation of Lebanese soil. On Friday, Hezbollah officials warned in a statement that if Israeli forces remained in Lebanon beyond Sunday, it would amount to “an attack on Lebanese sovereignty and the beginning of a new chapter of occupation.”

Some Hezbollah lawmakers have vowed retaliation. But other officials from Hezbollah — which has been militarily and politically battered in recent months — instead shifted responsibility for responding to Israel to the Lebanese government. The group’s statement on Friday said that it was up to the state “to reclaim the land and wrest it from the grip of the occupation.”

That shifting of responsibility is a tried-and-true tactic for Hezbollah, which just a few months ago had called on the state to provide for thousands of Lebanese displaced by a war it had pulled the country into. Still, the political posturing from a group whose founding principle is resisting Israeli occupation reflects Hezbollah’s current weakened state.

After 14 months of fighting, the Shiite Muslim group’s military ranks are battered, and its loyal support base is weary after months of displacement and destruction. Its patron Iran has also been weakened by Israel, casting doubt over Iran’s ability to provide millions of dollars for rebuilding the homes of Hezbollah supporters in Lebanon, as it did after Hezbollah’s monthlong war with Israel in 2006.

And in neighboring Syria, rebels toppled an Iran ally, the dictator Bashar al-Assad, cutting off Hezbollah’s land bridge for receiving weapons and cash from Iran.

These blows have loosened Hezbollah’s once iron grasp on political power in Lebanon, shifting the country’s political sands for the first time in decades. Earlier this month, Lebanese lawmakers elected a new president, Mr. Aoun, after years of political gridlock that many analysts had attributed to Hezbollah. Days later, lawmakers named Mr. Salam, a prominent diplomat whom Hezbollah had long opposed, as prime minister.

In a country where no major political decision had been made without Hezbollah’s blessing for years, those developments underscored just how much ground the group has lost.

But Middle East experts have warned against writing off Hezbollah’s political weight just yet. And if Israel continues to occupy Lebanon, it could revitalize the group’s mostly Shiite Muslim support base as it looks for a patron and protector against Israeli forces.

“I believe neither parties have an interest in resuming the war,” said Sami Nader, the director of the Institute of Political Science at Saint Joseph University of Beirut. “But as long as Israel is occupying Lebanon, it’s reviving the narrative of Hezbollah.”

Sara Chaito contributed reporting.

Some Australians were in no mood to celebrate the country’s national day on Sunday because they had long seen it as a reminder of colonial oppression. A few protesters took that antipathy a step further — by vandalizing statues to British settlers and an English king.

The damage done in Sydney, Melbourne and Canberra was a fresh sign that Australia Day, which commemorates when a British fleet sailed into Sydney Harbor to start a penal colony in the late 18th century, remains divisive.

Even as some Australians mark the holiday with barbecues and pool parties, critics note that it set in motion centuries of oppression of Indigenous people. Some prefer to call it Invasion Day or Survival Day, and they make their displeasure clear through protests or other actions.

In Sydney this week, a statue of Captain James Cook, who claimed part of the Australian continent for the British crown in 1770, was drenched in red paint. Its hand and nose were severed, too. The statue had been restored after facing a similar attack last year.

In Melbourne, a monument to John Batman, an explorer who settled the city on lands occupied by Aboriginal people, was toppled and destroyed early Saturday. Protesters in Melbourne also spray-painted the words “land back” on a memorial for Australian soldiers who died fighting in World War I.

And on Sunday in Canberra, the capital, there was graffiti on a statue of King George V. “The colony is falling,” someone had written on its base in red paint.

Australian officials condemned the vandalism.

“We should find it in our hears and in our minds to respect differences of views but not let it turn ugly,” said Jacinta Allan, the state premier of Victoria, according to a report by the television station 9News.

Representatives for the police in the states of Victoria and New South Wales said on Sunday afternoon that there had been no arrests or charges in connection with the vandalism in Sydney and Melbourne. The police in Canberra did not immediately respond to an inquiry.

People have protested Australia Day for decades. Recent protests were bolstered by the global Black Lives Matter movement, in which people in the United States, Britain and elsewhere toppled statues they saw as symbols of racism and oppression.

Last year in Melbourne, a Captain Cook statue was sawed off at the ankles, and a monument to King George V was beheaded.

Many Australian officials are keenly aware of their country’s racist colonial past, and they’re not afraid to say so publicly. In one example, the City of Melbourne’s website has a section on “truth-telling” that talks about developing “a shared understanding of the impacts of colonization and dispossession on Aboriginal peoples.”

But merely acknowledging historical wrongs is not enough for some Indigenous activists. That was clear when King Charles III visited Australia last year.

“You are not our king,” a voice rang out shortly after Charles, who retains the ceremonial title of head of state in the former British colony, finished addressing Parliament. “Give us our land back. Give us what you stole from us.”

The voice belonged to Lidia Thorpe, an Indigenous senator and activist for Aboriginal rights. As security guards hustled her out of the chamber, she accused British colonizers of genocide and demanded that Britain enter into a treaty with Australia’s Indigenous population.

The king watched impassively from the stage.

The family arrived at the ornately carved temple in western India bearing a special sweet of dried milk and clarified butter. It was a desperate offering for their son’s safety: He had just crossed into the United States, only days before President Trump took office promising a fierce crackdown on illegal immigration.

In their village in Gujarat, the home state of Prime Minister Narendra Modi, the markers of migration are everywhere. Plaques on buildings trumpet donations from Indians in America. Houses sit locked and empty, their owners now in the United States — many legally, many not.

Mr. Trump’s threats of mass deportations of illegal immigrants have raised the loudest alarms in countries closer to the United States, like in Mexico and Central America. But the fear and uncertainty — and the potential for political repercussions — are also rippling through India.

India is one of the top sources of illegal immigration to the United States, according to the Pew Research Center. As of 2022, more than 700,000 undocumented Indians were living in the United States, the center estimates, making them the third-largest group, behind Mexicans and Hondurans.

Some Indians arrive legally and overstay their visas. Others cross the borders without authorization: In 2023 alone, about 90,000 Indians were arrested as they tried to enter the United States illegally, according to U.S. government data.

India’s government, which has expanded defense, technology and trade ties with the United States, has expressed confidence that it is better positioned than most to weather the global reckoning with another “America First” administration. Mr. Modi has a bond with Mr. Trump, calling him “my dear friend” as he congratulated him on taking office for a second time.

Nevertheless, there are signs that India is trying to keep Mr. Trump on its good side by cooperating with his clampdown on illegal migration.

Indian news outlets reported this past week that the government had been working with the new administration to take back 18,000 Indian illegal immigrants who are under so-called final removal orders.

According to those reports, India’s goal is to protect its legal pathways for immigration to the United States, like skilled-worker visas, and avoid the punitive tariffs Mr. Trump has threatened to impose over illegal migration. Helping his administration could also spare India the embarrassment of being caught up in the publicity of Mr. Trump’s crackdown.

Indian officials would not confirm the specifics of the news reports to The New York Times. But they noted that deportations from the United States to India were not new — more than 1,000 Indians were sent back last year — and said that they were working with the Trump administration.

“Our position is that we are against illegal migration,” said Randhir Jaiswal, the spokesman for India’s Foreign Ministry. “We have been engaging with U.S. authorities on curbing illegal immigration, with the view of creating more avenues for legal migration from India to the U.S.”

Those legal routes — namely, H-1B visas for skilled workers and visas for students — have been a subject of heated debate among Mr. Trump’s supporters. Elon Musk and other tech moguls say the H-1B visas are needed to recruit the best talent to the United States. More nationalist voices say the jobs filled by those visa holders should go to Americans.

The State Department said the Trump administration was working with India to “address concerns related to irregular migration.” The new secretary of state, Marco Rubio, held his first bilateral meeting on Tuesday with India’s foreign minister, S. Jaishankar — an indication of the growing importance of the U.S.-India relationship.

The intensified focus on migration is politically sensitive in India.

Mr. Modi, the country’s most powerful leader in decades, has cast himself as a driving force behind economic growth that he says will eventually make India a developed nation. But his own home state, Gujarat, once hailed as an economic miracle under his leadership, is one of India’s largest sources of illegal migration to the United States, according to police officials.

Though Washington is looking to India as an alternative to China in global industrial dominance, its uneven economy — by some measures, one of the most unequal in the world — still impels large numbers of Indians to take enormous risks to make it to the United States.

In the Mehsana district of Gujarat, almost every family has a member in the United States, legally or illegally. Some return only for annual visits to see aunts and uncles. Mehsana is frequently in the news, with reports of its migrants dying while trying to climb a border wall into the United States, reach its shores by boat or make their way over the frozen northern border during winter.

Migration to the United States has traditionally been a status symbol among Gujaratis. Families who have no members in the United States have trouble matching their children in marriages, said Jagdish, 55, a worker at the local college in the village of Jasalpur whose son and daughter-in-law are in the United States illegally.

Jagdish, who asked that his last name not be used, said his son had spent five months in Mexico waiting to cross the border five years ago. Upon entering the United States, he was jailed for three months before being released. He now works at a cafe there, and his wife joined him last year.

It cost the family more than $70,000 to get them to the United States — a mix of “hard-earned money, my life’s savings” and loans, Jagdish said.

“I don’t buy new clothes, I have cut down on fruits and milk,” he said. “I need to repay the loans.”

Outside the village temple, a husband and wife who run a Subway franchise in the United States, where they have lived for two decades, were on their once-a-year visit. The husband, Rajanikant Patel, tried to offer some reassurance about Mr. Trump, couched in the “no one knows” air that characterizes much talk about the new administration.

“Trump will do what he has to do,” Mr. Patel said. “But Trump needs people to work there. We are laborers there. It’s such a huge country. Who will work and manage there?”

Indians began moving to the United States in large numbers in the 1960s, when India was among the world’s poorest nations and American immigration policy was easing.

The pull is strong even today, with India now the world’s fifth-largest economy. Given its immense inequality, economic growth has not necessarily translated into better services or higher standards of living for most.

“The quality of life here and there cannot be compared,” said Mr. Patel’s wife, Nila Ben.

Immigration consultants said they had seen a decline in visitors as word spread that it was becoming harder to enter the United States, a tightening that began during the Biden administration and that Mr. Trump is moving to drastically increase.

Varun Sharma, the director of an immigration consultancy, said about half of his potential clients inquired about illegal routes into the United States. He politely turns them down, he said.

Many undocumented immigrants now come from the new middle class. In some cases, Indians who arrive on student visas stay past the expiration date. In other cases, migrants first fly to a third country on a visitor’s visa, then slowly make their way to the United States by land or sea.

Vishnu Bhai Patel, a lemon trader from a nearby village, said he hoped that Mr. Trump “shows some leniency for divided families like mine — half of the family is here and half there.” He said he hoped that his daughter, who is studying engineering in the United States, could stay on after graduating and then invite him to come legally, too.

“My dream is for her to never come back,” he said.

Mujib Mashal contributed reporting from New Delhi.

Outside a warehouse in southern Gaza one day this week, a small crowd of men and boys waited their turn for a bit of the humanitarian aid that Gaza — sick, starving, freezing Gaza — has desperately needed. They walked away with sacks of flour and cardboard boxes of food, many dragging their precious cargo behind them in two-wheeled shopping carts.

It was an orderly sight that had become rare in the territory since the war began more than 15 months ago. Israeli restrictions on aid, a security collapse that allowed widespread looting of aid trucks and other obstacles had combined to limit the food, water, tents, medicine and fuel that reached civilians amid an Israeli siege on the strip.

In the week since a cease-fire agreement stopped the fighting in Gaza, Palestinians in Gaza and aid officials say that more food deliveries and other much-needed items are streaming in. The question now is how to maintain the level of aid they say Gaza needs, despite many logistical challenges and uncertainties over how long the truce will hold.

The United Nations moved as much food into Gaza in three days this week as it did in the entire month of October, the interim head of the U.N. humanitarian office for Gaza, Jonathan Whittall, said in a briefing on Thursday.

Other U.N. agencies and aid groups were distributing medical supplies and fuel to power hospitals and water wells, among other types of assistance, and helping to repair critical infrastructure. Tents were set to enter soon, and bakeries were expected to start supplying bread by Friday, according to the United Nations.

Since the start of the cease-fire, civilian police officers belonging to the Hamas government have re-emerged, which appears to have restored some security and order to the enclave. The show of Hamas control, however, may complicate prospects for a durable peace in Gaza.

COGAT, the Israeli government agency that oversees policy in Gaza and the West Bank, did not respond to a request for comment, but it said in a post on social media on Friday that 4,200 aid trucks had entered the Gaza Strip over the past week after being inspected.

Throughout the war, Israel said that it was not limiting aid into Gaza and blamed humanitarian agencies for failing to distribute the supplies it admitted into the enclave after screening.

In all, anywhere between about 600 and 900 truckloads of aid have arrived in Gaza each day since the cease-fire took effect on Jan. 19, dwarfing the few dozen trucks that had been entering daily in recent months.

By Tuesday, Kholoud al-Shanna, 43, and her family had received a bag of flour from the World Food Program, the first in two months.

It was welcome. But “we’re still missing the basics,” Ms. al-Shanna said. “My kids haven’t had fresh vegetables in so long that they’ve almost forgotten what they taste like. How are we supposed to survive on just flour?”

Improvements were coming on that front, too. Before the war, Gaza was supplied with a mix of donated aid and goods for sale. Small amounts of imported fresh produce, meat and other food continued to be sold in markets until Israel banned most commercial items late last year, arguing that Hamas was profiting off the trade. Some commercial goods have entered Gaza this week, according to aid workers, bringing fresh vegetables and even chocolate bars to markets at lower prices than shoppers have seen in many months.

Distributing the aid once it enters Gaza remains a work in progress. Many roads are in ruins after 15 months of war, though Gaza municipalities are starting to clear debris. Unexploded ordnance still litters the enclave, making distribution and repairs dangerous.

About 500 trucks carrying a mix of aid and commercial goods entered Gaza each day before the war. The cease-fire agreement envisions 600 trucks entering each day, which aid officials say they will be hard-pressed to sustain on their own.

“It cannot be delivered just by the United Nations, no way,” Philippe Lazzarini, the head of the United Nations Relief and Works Agency, or UNRWA, the primary lifeline for Palestinian refugees, said days before the cease-fire took effect.

UNRWA’s precarious situation is another potential hindrance: While U.N. officials say the agency is crucial to the aid effort because it forms the backbone of supply chains and services in Gaza, Israel has moved to ban the agency over accusations that it shielded Hamas militants. Aid officials say there is nothing comparable to take its place.

The biggest challenge of all is the sheer scale of the emergency. Though aid may be rolling in now, aid officials said, Gaza has been so lacking in assistance that it will take a deluge of supplies just to stabilize the population and prevent more deaths, to say nothing of eventual reconstruction.

Gaza will also need educational and psychological services and other support to begin to recover, officials say.

The number of trucks recently entering Gaza “is still a drop in the ocean compared to the amount of aid needed to catch up on what has been a massive dearth over the last year and a half,” said Bob Kitchen, the vice president for emergencies at the International Rescue Committee.

Some obstacles are gradually yielding. Israel’s evident willingness to usher in a surge of aid has resolved what aid officials and governments that donated assistance say was the biggest hurdle to getting Gaza what it needed. Saying its goal was to keep Hamas from resupplying through aid shipments, Israel had imposed stringent inspections on the assistance entering Gaza and restricted its movement once inside Gaza, frequently delaying or outright stopping delivery.

Aid workers no longer need to ask permission from the Israeli military to move around Gaza, except from south to north, speeding up the process. Before the cease-fire, many trucks designated to ferry aid to warehouses around the strip sat paralyzed for lack of fuel; now fuel is entering.

Israel still prohibits agencies from bringing in a long list of items that aid officials say are vital to the emergency response but that Israel deems “dual use,” meaning they could also be used by Hamas for military purposes. That has included everything from scissors to tent materials.

Some of those restrictions have been lifted, however, aid officials say, and talks are continuing about lifting more.

Another problem plaguing aid distribution in Gaza for months was looting, which diverted much of the aid meant for civilians.

The situation in Gaza deteriorated after the Israeli military invaded Rafah, in southern Gaza, in May, seeking to oust Hamas from what Israel said was one of its final strongholds. Hamas’s security forces fled, and organized gangs — with no one stopping them — began intercepting aid trucks after they crossed into Gaza.

International aid workers accused Israel of ignoring the problem and allowing looters to act with impunity. The United Nations does not allow Israeli soldiers to protect aid convoys, fearing that would compromise its neutrality, and its officials called on Israel to allow the Gaza police, which are under Hamas’s authority, to secure their convoys.

Israel, which has sought to destroy Hamas in Gaza, accused it of stealing aid and said the police were part of its apparatus. In the end, security broke down so badly that many aid groups kept their deliveries sitting at Gaza’s borders rather than risk the dangerous drive into Gaza.

But fears that organized looting would continue after the cease-fire have eased. Policemen are once again patrolling much of Gaza. While some people are still pulling boxes from trucks — scenes described by aid officials and witnessed by a New York Times reporter — it is now on a far smaller scale.

Palestinians in Gaza say that as aid becomes more widely available, people will have less incentive to loot.

“I’ve noticed a clear improvement — more people are getting food parcels today,” said Rami Abu Sharkh, 44, an accountant from Gaza City who had been displaced to southern Gaza. “I hope it continues until theft is eliminated completely.”

Farnaz Fassihi contributed reporting from New York.