More migrant buses have already begun arriving in cities far from the border.

More than a week before the anticipated lifting of Title 42, a bus pulled into the Port Authority bus terminal in midtown Manhattan at 7 a.m. sharp, carrying 41 new migrants.

Most were families with children, and most were from Venezuela, said Ilze Thielmann from Team TLC NYC, a group of volunteers who offer support to the new arrivals. A second bus arrived at around 1 p.m. with 50 people on board, the group said.

A year after the first increase in migrants in New York City, the bus arrivals at Port Authority have become a tightly controlled operation: Arriving migrants are greeted by volunteers, then whisked away to centers organized by the city to connect them to services and homeless shelters, often run out of hotels and other centers.

The buses slowed earlier this year, but now, the city is bracing for many more.

The Texas governor, Greg Abbott, a Republican, has resumed a program that started last year to send migrants to cities controlled by Democrats, including Los Angeles, Chicago, Denver and Washington, D.C.

Manuel Castro, New York City’s immigration commissioner, said that up to 1,000 migrants could arrive daily in the coming weeks.

Last week, Mr. Castro told reporters at the bus terminal that Governor Abbott was “clearly using them for political purposes,” echoing Mayor Eric Adams, who has been in a rhetorical war with the Texas leader since last summer. The mayor has also criticized President Biden for not doing more to help cities cope.

“It’s becoming more of a crisis, because the federal government refuses to intervene and stop what Governor Abbott is doing, to provide immediate relief to people,” added Mr. Castro.

For his part, Mr. Abbott said last week that his state would continue “this necessary program” to provide relief to border cities.

Chicago has faced a similar strain, and a similar dilemma, as New York: Many residents embrace the city’s reputation as a sanctuary for new immigrants, but the arrival of more than 8,000 migrants has tested its resolve. (Nearly 60,000 have come to New York, although not all have stayed.)

In Washington, D.C., an increasing number of migrants could exacerbate an already difficult situation for organizations still trying to arrange housing and services for those bused to the city last year, said Diana Fula from the Congregation Action Network. Ms. Fula said hotels that have been provided for temporary shelter are full.

“We will end up seeing migrants sleeping in tents or next to cars because this is how the government is treating migrants who don’t have any other option; this will soon be our reality in the city,” she said in a phone interview.