Since arriving from Venezuela four months ago, Joana Rivas has slowly found some semblance of stability in New York City, picking up occasional cleaning jobs and enrolling her 9-year-old daughter at a public school in Manhattan.

As she navigates her new city, a crucial anchor for Ms. Rivas has been the free housing she was given at a hotel-turned-shelter near Times Square. On Tuesday, however, her time at the shelter ran out. Ms. Rivas had to keep her daughter home from school and pack their belongings to go and apply for new housing.

“Tonight, I don’t know where we’ll go,” Ms. Rivas, 39, said outside a welcoming center for migrants in Midtown Manhattan. “I came here just to see what they would tell me, with the hope that my daughter has somewhere to stay tonight.”

New York City has begun to evict dozens of migrant families that had reached their 60-day limits on stays in the homeless shelter system, the latest effort by the city to urge more of them to leave and find permanent housing. Nearly 70,000 migrants are living in a patchwork of hotels, homeless shelters and giant, winterized tents set up by the city.

The first wave of evictions coincided with an unexpected and significant hurdle. City officials announced that they would temporarily evacuate 1,900 migrants currently housed in a tented shelter site in southeast Brooklyn because of a rainstorm headed for the city on Tuesday night.

A spokeswoman for City Hall said that the relocation of migrant families living in the tent dormitories at Floyd Bennett Field, on the shores of Jamaica Bay, was taken “to ensure the safety and well-being of individuals working and living at the center,” which is an area that flooded during the last major storm. The families would be bused to a Brooklyn high school on Tuesday afternoon, city officials said.

The migrant families with children forced out of the city’s buckling shelter system as a result of the 60-day limit will be allowed to reapply for another spot. But they will have to leave the shelters they have called home for weeks and sometimes months and go to a city intake center with their suitcases to be assigned new beds, potentially at a different location.

The reshuffling, in the dead of winter, has been attacked by critics of Mayor Eric Adams as unnecessarily disruptive for migrants still finding their grounding in a new city. Advocates have also raised concerns for migrant children who might end up in living arrangements that are farther away from the public schools they are enrolled in.

Mayor Adams defended his decision to force migrant families to reapply for shelter, arguing at a news conference on Monday that the city was being overwhelmed by new arrivals and that families living in hotels had to find more stable places to live. He promised, as he has before, that no families with children would be forced to sleep on the streets.

“We’re doing it in a very humane way,” he said.

Since last spring, nearly 170,000 migrants have traveled from the southern border to ask for help in New York City, many of them drawn to the city’s unique legal mandate to provide free housing to anyone who asks for a bed.

The Adams administration has opened more than 200 emergency shelters to help house migrants, most of whom are no longer in the city’s care. But the city has also begun limiting the number of days migrants can stay in shelters to free up beds and coax those in shelters out of the city’s care.

About 4,400 total families have been given 60-day notices that will take effect in the coming weeks, officials said. They said that case workers have met with the families to discuss potential living arrangements. The first migrants impacted by the 60-day limit on Tuesday were 40 migrant families staying at the Row, the Times Square hotel where Ms. Rivas was staying.

The families were supposed to check out from the shelter on Tuesday morning, officials said. Those who wanted to reapply for shelter had to go to the Roosevelt Hotel, a Midtown hotel about a 15-minute walk away that has been transformed into the city’s main migrant intake center.

Dr. Ted Long, a senior vice president at the city’s public hospital system who is helping oversee the effort, said that the city would prioritize helping migrant families with young children to find new housing near their schools and to minimize disruption.

“We’re going to move heaven and earth to place them in a new hotel as quickly as possible,” he said on Monday.

Despite the reassurances, Democratic officials to the political left of Mr. Adams quickly attacked the displacement of migrant families, using stern language to call on the mayor to suspend the time limits placed on shelter stays. At a rally in front of City Hall on Monday, Brad Lander, the city comptroller, described the evictions of families as “one of the cruelest things the city has done in generations.”

“We will not throw people out of warm shelters in cold winters,” he said. “We will not allow this to be a city that displaces kids from their public schools in the middle of the school year.”

A group of advocacy organizations also denounced the 60-day limits as harsh and destabilizing for migrants. The potential disruption to schooling could have disastrous effects on children, some educators have said, and advocates said the change in home addresses could disrupt migrants’ access to their mail, including important immigration court notices.

“Instead of focusing on school, work, and building their lives here, newly arrived immigrants will have to navigate increased bureaucratic hurdles to maintain shelter and access immigration assistance,” the New York Legal Assistance Group said in a statement.

Migrant families who said they had lived at the Row — at least one for 14 months — began to arrive at the Roosevelt Hotel seeking new shelter assignments, some of them with suitcases. Some brought their children with them, deciding against sending them to school on Tuesday without knowing where they would sleep by day’s end.

Anthony Gomez and his wife, who is pregnant, said they woke up on Monday to banging on their door notifying them they would have to leave the hotel the following day — their 60th day in New York after coming from Venezuela.

“We hadn’t realized it was time,” said Mr. Gomez, who is 29 and has been working as a deliveryman for Uber. “They knocked on our door like they were police. We don’t know what happens now, we have no clue.”

Other migrant families said they had plans to leave the shelter system, and even New York City, for good.

Angel Gonzalez, 36, luggage in tow, left the Row hotel for the last time on Tuesday morning. He had lived there with his family, including two children, ages 13 and 15, for 11 months after migrating from Venezuela. He said the free housing allowed him to save some money and “get everything sorted” to move to Philadelphia, where he has a family member.

“I’m grateful for New York, for this country and thankful for everything,” Mr. Gonzalez said on his way to catch the train. “This is a new opportunity.”

Single adults were already subject to shorter 30-day limits late last year, prompting long lines outside an East Village school where dozens of migrants slept and stood in the freezing cold in December, hoping for a chance to reapply for another bed.

City officials have said that the time limits on shelters stays, which have also been enacted in other cities grappling with an influx of migrants, including Chicago and Denver, are helping migrants leave the shelter system permanently.

The wintertime reshuffling of families comes as Mayor Adams continues to expand his efforts to curb the flow of migrants into the city. The mayor, who traveled to Latin America last year to personally dissuade migrants from coming to New York City, is offering migrants free bus and plane tickets out of New York.

Last month, the mayor also escalated his feud with Gov. Greg Abbott of Texas, a Republican, who has sent more than 30,000 migrants in Texas-paid charter buses to New York City since last year. On Dec. 27, Mr. Adams imposed new rules and penalties on charter buses transporting migrants to New York City, limiting the hours and locations at which they can drop them off.

Last week, the Adams administration filed a lawsuit against 17 bus companies that seeks about $700 million in damages, the amount the city says it has spent to shelter the migrants the companies have bused to New York.

Emma Fitzsimmons and Liset Cruz contributed reporting.