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At least 68 migrants arrived in NYC over the weekend on buses sent by Texas Gov. Abbott

(CNN) New York Mayor Eric Adams has claimed that some migrants are being "forced" on buses from Texas, as 14 more asylum seekers arrived in the city Sunday on another bus sent by Texas Gov. Greg Abbott.

Fifty-four asylum seekers arrived in New York Friday on board a bus from Texas, according to the Mayor's Office of Immigrant Affairs.

Abbott's office indicated that New York is now a designated "drop-off location for the busing strategy as part of the governor's response to the Biden administration's open border policies overwhelming Texas communities," according to a statement released Friday announcing the arrival of the first bus.

"It's unimaginable, what the governor in Texas has done," Adams told reporters Sunday. "When you think about this country, a country that has always been open to those who were fleeing persecution and other intolerable conditions, we've always welcomed that. And this governor is not doing that in Texas, but we are going to set the right message, the right tone, of being here for these families."

On Monday, Adams called it a "mean and cruel thing."

Abbott claimed in a statement Monday, though, that the asylum seekers freely chose to go to New York, "having signed a voluntary consent waiver, available in multiple languages, upon boarding that they agreed on the destination."

"What's horrific is the thousands of illegal immigrants overrunning and overwhelming our border communities with populations smaller than a New York City borough, and Mayor Adams is hypocritically upset about welcoming a few dozen into his sanctuary city," Abbott said. "If the mayor wants a solution to this crisis, he should call on President Biden to take immediate action to secure the border—something the President continues failing to do."

A fierce critic of the Biden administration's immigration policies, Abbott began sending hundreds of willing migrants on buses to Washington, DC, earlier this year as an affront to the administration. Abbott's office has said that "to board a bus or flight, a migrant must volunteer to be transported and show documentation from DHS."

More than 5,100 migrants have arrived in Washington from Texas on more than 135 buses, according to the governor's office.

"In addition to Washington, D.C., New York City is the ideal destination for these migrants, who can receive the abundance of city services and housing that Mayor Eric Adams has boasted about within the sanctuary city," Abbott said Friday.

His state has been busing people to Washington, DC and New York City "to provide relief to our local partners," Abbott said in the statement.

Manuel Castro, commissioner of the mayor's immigrant affairs office, told CNN New York is a right-to-shelter city, so anyone who needs shelter may receive it. However, the city has "exhausted" its regular shelter space, so it has leased additional space at hotels, he said.

"These are families, these are people," Castro said. "They have a right to be here as asylum-seekers and New York is here to welcome them. They frankly need a lot of support. They've traveled a long way to get here."

Adams repeated Monday he hoped to be in touch directly with the White House about federal assistance on the matter.

"Right now, we are at a state where we must get to the assistance from the federal government," Adams said.

The Biden administration has "been in regular contact with Mayor Adams," White House spokesperson Abdullah Hasan said, and the administration is "committed to working with them as we do effectively with other local leaders through FEMA funding and other support."

"As we have always said, there is a process in place for managing migration flows, and Republicans governors should stop meddling in that process and using desperate migrants as political tools," Hasan said.

Some migrants did not want to come to New York, mayor says

Around 40 people were expected to arrive Sunday, Adams said, adding that it is not known if others got off the bus before it arrived in New York.

Some of the migrants who did arrive Sunday told the mayor they had relatives in other cities and did not plan to come to New York.

Some families "wanted to go to other locations and they were not allowed to do so," Adams said Sunday. "They were forced on the bus with the understanding that they were going to other locations that they wanted to go to, and when they tried to explain, they were not allowed to do so."

Families whose desired destination was not New York said it took three days to get to the city from Texas, according to Castro, who also met with them, noting many "were hungry and thirsty, with small children."

Adams issued an emergency declaration last week to "rapidly procure shelter and other services" for people seeking asylum in the city. Those who want to stay in New York will go to shelters, and volunteer groups will be helping "those who want to go somewhere else" to travel to another location, the mayor said.

New York Taxi Workers Alliance executive director Bhairavi Desai told CNN they had 19 yellow cabs and Uber cars ready to voluntarily transport refugees arriving from Texas from the bus stop to shelters or to homes of friends or family in the area Sunday morning.

"Everyone we saw looked exhausted by the journey. Many people looked relieved to finally reach safety but there were also people crying from the trauma," Desai said.

"It's wrong, and in our melting pot city, it won't work," Desai added. "We wanted to do our part to let new neighbors know that they're welcome here, and in NYC there are people who will help."

Around 4,000 asylum seekers have entered the New York City shelter system since late May, according to Adams, who said the influx is the primary driver of around a 10% increase in the city's Department of Homeless Services' census. New York's shelter system is currently receiving more than 100 asylum seekers looking for some form of housing per day, on average, according to the homeless service's department.

CNN has asked the mayor's office for the total number of migrants the city is processing from Texas.

Generally, once migrants are processed by federal authorities and released from custody, they are allowed to move throughout the country while they go through immigration court proceedings. They are often released in Texas and other border states, and then continue on journeys to other parts of the country.

An immigration judge will ultimately decide if they are allowed to remain in the US or be deported.

CNN's Laura Ly, Polo Sandoval, Kristina Sgueglia, Mark Morales, Priscilla Alvarez and Maria Santana contributed to this report.