SAN ANTONIO — In Houston, nearly 2,000 Haitian migrants have arrived this week from the small border community of Del Rio, with buses pulling up to a huge shelter nearly every hour. In San Antonio, hundreds more have been allowed by the U.S. authorities onto flights to destinations as far away as New York, Boston and Miami, paperwork in their pockets permitting them to remain in the country.

Immigration and Customs Enforcement has deported about 2,000 migrants in recent days on chartered flights to Haiti as the Biden administration tries to deter more people from rushing to the border. But the authorities have also permitted thousands more to travel to cities across America, where they may live for months or years as they await immigration hearings.

“We are so happy to be in America,” said Inso Isaac, 40, who left Haiti years ago and was living in Chile until he, his wife and their 2-year-old son made the dangerous journey across several countries and arrived last week in Del Rio. On Wednesday, the family boarded a flight to New York, where they planned to stay with relatives on Long Island. “We want to start a new life here,” he said.

A chance to settle in the United States, however slim, has driven the latest surge, compelling more than 14,000 migrants to wade across the Rio Grande and into Del Rio over the past week, where they have encountered armed National Guard troops and Border Patrol agents on horseback. On Thursday, about 3,100 remained huddled in squalid conditions under the international bridge that connects Del Rio to Mexico, circumstances that have prompted outrage from both Republicans and Democrats.

Images of the agents on horseback rounding up migrants and of dozens of state police vehicles blocking entrance across the river have fueled criticism from Democratic lawmakers and administration officials that the Haitians are being treated inhumanely. On Thursday, the Department of Homeland Security said the horse patrol unit in Del Rio had been temporarily suspended and that the agents’ actions were being investigated. The U.S. special envoy to Haiti has also resigned in protest of mass deportations, two officials said, and sent a blistering letter to Secretary of State Antony J. Blinken.

“I will not be associated with the United States’ inhumane, counterproductive decision to deport thousands of Haitian refugees and illegal immigrants to Haiti, a country where American officials are confined to secure compounds because of the danger posed by armed gangs to daily life,” Daniel Foote, who was appointed to the position in July, wrote in a letter dated Wednesday.

Jen Psaki, the White House press secretary, said at a news briefing on Thursday that officials had aimed to rapidly turn away single adults and migrant families. But some groups, including pregnant women and families with young children, have been allowed to remain in the United States because some countries accepting the deportees will not accept migrant families with young, vulnerable children.

Ms. Psaki said that the White House had been “horrified” by the images of the agents on horses rounding up migrants and that Mr. Biden, whose administration has faced the highest level of border crossings in decades, was working to develop a “humane” immigration system.

Still, criticism from immigration advocates continued building on Thursday over the decisions as to who could stay and who could not. More than two in three Haitian migrants who have been expelled from the border and returned to Port-au-Prince, Haiti’s capital, are women and children, according to initial estimates from UNICEF. Meanwhile, conservatives criticized the Biden administration for admitting so many others.

It was unclear on Thursday how many Haitians had been deported as opposed to being allowed to enter the country and await asylum hearings. All told, nearly 2,000 Haitians had been returned to Haiti by late Thursday, about 40 percent of them family units, according to a Department of Homeland Security official. Resettlement groups said they were aware of about a roughly equal number from Del Rio who had been given permission to stay in the country.

Under President Donald J. Trump, the asylum system was essentially brought to a halt, as almost no migrants were allowed to enter the country while their claims for protection were heard; instead, they were required to remain in Mexico, often settling in decrepit camps near the border. By contrast, the Biden administration has allowed more to enter, and remain in, the United States while their asylum cases unfold.

But because the immigration courts are severely backed up, the process can take several years, allowing people to effectively settle in the United States. If they lose their cases or do not attend their court hearings and remain in the country illegally, they would be joining millions of undocumented immigrants already living in the shadows.

At the San Antonio airport on Wednesday evening, a number of Haitian families who had been in Del Rio waited to board planes to various American cities. Mr. Isaac, holding a paper that instructed him to report to an Immigration and Customs Enforcement office within 15 days of arriving in New York, said he would need to find a lawyer. But that felt like an easy task after spending $12,000 and several weeks traversing South and Central America by foot and bus and swimming in dirty rivers.

Nearby, Israel Fleurios, 31, and Widna Azema, 35, waited for a flight to Miami, where they planned to stay with an aunt of Mr. Fleurios’s. The couple fled Haiti five years ago and had spent several years in Brazil before making the journey to Del Rio, an unlikely spot for Haitian migrants but a border crossing that they had heard was accessible.

Ms. Azema was pregnant when they left Brazil, and she gave birth to a daughter, Bruna, in Guatemala, and carried her the rest of the distance to Texas. The couple has another daughter, Valentina, 3, who has a skeletal disorder that prevents her from walking upright.

“I think they let me through because they saw how bad she was,” Ms. Azema said of the border authorities’ reaction to Valentina’s condition. “Everybody with children like us were allowed to get through. We are appreciative.”

In a corner of the airport, many other Haitian migrants sat anxiously with their few belongings. Duperval Marie Ange, 42, watched her 5-year-old son, Mike, run around the terminal while they waited to board a flight to Fort Lauderdale, Fla. Like the others, she had spent a grueling week under the International Bridge in Del Rio. She had cringed when she saw images of the agents on horses chasing migrants. The photos were troubling, she said in broken Spanish she had learned while living in Chile, but they also made her grateful to have been allowed into the United States.

“I cannot say anything bad,” she said. “La policia me ayudo. The police helped us. They gave us food. They let us cross. We are here.”

In Houston, hundred of Haitians were taken in at shelters. At one site, about 300 people were arriving every day this week, said Carlos Villarreal, an elder with the Church of Jesus Christ of Latter-day Saints, which operates that shelter. His shelter was only receiving families, he said, and many of them included children or pregnant women.

“At least 25 percent of the families include pregnant women,” Mr. Villarreal said. “Some of them have been traveling for weeks from South America in extremely challenging conditions.”

Families are tested for Covid-19 upon arriving at the Houston shelter and are then given food, water and a change of underwear, in addition to access to showers and beds.

“Some of our families have been traveling for weeks, not bathing, not eating properly, without access to feminine hygiene products,” Mr. Villarreal said.

From Houston, many of the families, who typically stay less than 24 hours at Mr. Villarreal’s shelter, then travel to places around the United States where their relatives are living. The relatives are expected to pay for the airfare, but volunteers have mobilized to do so if that isn’t possible.

Mr. Isaac, his young son in his arms, left Haiti in 2017, fleeing what he said was a never-ending cycle of violence, poverty and natural disasters. In Chile, he met his wife, and they had a son, Hans, who was burned in an accident and requires medical attention. There, he worked in construction, hotels and restaurants, and he said he planned to look for similar jobs in New York.

He and his wife had not planned it this way, but Hans celebrated his second birthday on Wednesday — in an American airport that was their near the end of what had been a very long journey.

“I felt bad, because we all came here for the same reasons,” he said of the many Haitian migrants he met in Del Rio. “I knew not everyone was going to make it through. We were lucky.”