The U.S. has deported more than 2,000 Haitians, including some families, from a Texas camp with plans to deport more.

The Biden administration has deported more than 2,000 Haitians since last week, a nearly even mix of single adults and families, out of the more than 12,000 that have been apprehended in a small Texas town by border officials, according to internal accounts.

Another 3,900 Haitians are in government custody and will be deported or put into removal proceedings, the Department of Homeland Security said in a statement Thursday night.

Nearly 4,000 of the migrants have been released with instructions to report to immigration officials, and thousands more are in the process of being interviewed, according to an official familiar with the information who was not authorized to discuss it publicly and spoke on condition of anonymity. Jen Psaki, the White House press secretary, said during a news briefing on Thursday that officials had intended to rapidly turn away all of the single Haitian adults and migrant families who arrived since last week after crossing the Rio Grande into Del Rio, Texas.

But some migrant families, she said, have been allowed to stay, including some families with young children, because some authorities in Mexico would not accept migrant families with young, vulnerable children. Limited shelter capacity in Mexico has also hindered the administration from turning away some families, she said.

The Department of Homeland Security said about two-thirds of the Haitian migrants that had arrived were traveling in family groups. The department did not immediately respond to questions about who has been deported.

The agency said that as of Thursday, about 4,000 migrants were still waiting under a bridge in Del Rio, where at one point, 15,000 migrants were crammed together in squalid conditions, raising concerns of a potential humanitarian crisis.

The Biden administration’s response to the spike in migrants has drawn condemnations from immigration and human rights advocates, as well as members of the president’s own party. The senior U.S. envoy for Haiti policy resigned on Thursday over what he called the administration’s “inhumane, counterproductive decision” to send Haitian migrants back to a country that has been racked this summer by deadly natural disasters and political turmoil.

Images of Border Patrol agents on horseback pushing back Haitian migrants crossing the Rio Grande has sparked outrage, including by President Biden and Vice President Kamala Harris. The agents in question are under investigation for possibly mistreating migrants, and the horse patrol in Del Rio has been temporarily suspended, the Homeland Security Department said.

“Many of these vulnerable Haitian families have braved unforgiving terrain to reach our southern border and lawfully seek asylum,” Krish O’Mara Vignarajah, the president and chief executive of the Lutheran Immigration and Refugee Service, a refugee-resettlement agency, said in a statement on Thursday. “Returning them home, or to third countries with no infrastructure to support them, is as dangerous as it is unconscionable. To do so without so much as an interview or court hearing is downright un-American.”

Ms. Psaki said the president was working to develop a “humane” immigration system, “but we’ve also reiterated that it is our objective to continue to implement what is law, and what our laws are and that includes border restrictions.”

Zolan Kanno-Youngs contributed reporting.