WASHINGTON — Mirard Joseph was bringing food to his wife and child, he said, when a Border Patrol agent on horseback in Del Rio, Texas, “lashed at” him with his reins and dragged him by the collar toward the Rio Grande, which in that region separates the United States from Mexico.

The treatment of Mr. Joseph, an undocumented immigrant who was seeking asylum but has been deported, and other Haitian migrants by agents in September is the subject of a federal investigation after President Biden described it as “outrageous” and promised that “there will be consequences.” The union representing the Border Patrol agents has defended them, saying they were merely doing their job as thousands of migrants crossed into the small border town.

The migrants’ side of the story was laid out for the first time on Monday, in a lawsuit against the government filed in U.S. District Court for the District of Columbia. By Mr. Joseph’s telling, the agent in a widely published photograph let go of him when the horse appeared close to trampling him.

“It was the most humiliating experience of my life,” he said in the lawsuit, which recounts his experience and those of 10 other Haitians in Del Rio and during their expulsions. The complaint accuses the government of physical and verbal abuse, inhumane treatment and denial of due process under a public health rule that gives border officials authority to expel most people who cross into the United States illegally during the coronavirus pandemic. The migrants are being represented by the Justice Action Center, an immigrant advocacy organization.

The lawsuit alleges that the Biden administration knew an influx of migrants was coming but deliberately made no humanitarian preparations — a strategic decision, the suit says, meant to deter more Haitians from trying to cross into the United States. Among other things, the plaintiffs are asking to be allowed to return to the United States and remain there while they request asylum.

The photographs of Mr. Joseph and others in Del Rio prompted criticism of the Biden administration’s response to the thousands of Black migrants who were crossing illegally into the United States at the time, many in search of asylum.

Regardless of the outcome of the suit, the plaintiffs’ accounts could become part of the investigation into the agents’ actions in Del Rio, which is being conducted by Customs and Border Protection’s Office of Professional Responsibility. The Homeland Security Department said the office was reviewing images for possible misconduct and interviewing witnesses, Border Patrol employees and senior leaders at Customs and Border Protection.

Yet some have questioned how witnesses could be interviewed if so many of the migrants who had gathered in Del Rio were expelled to Haiti or returned to Mexico.

In Mr. Joseph’s case, the photographs prompted accusations that the agents had used their reins as whips — a claim that is central to the investigation. According to Mr. Joseph’s complaint, that was indeed what happened. But the Border Patrol union has said the agents were twirling their reins, as they are trained to do, to deter people from getting too close to their horses and therefore avoid injuries.

Several plaintiffs said they “saw officers on horseback using reins as whips against people” in the Rio Grande. One migrant, identified in the lawsuit as Esther, said agents on horseback chased her back into the river in the direction of Mexico. She said the horses nearly ran over her as the agents shouted, “Go back to Mexico.”

Another plaintiff, identified as Paul, said he saw Border Patrol agents beating Black migrants. As he was in the middle of the Rio Grande, Paul said, he saw agents cut a rope that had been strung across the river to help people cross more safely. He said he watched Haitians in deeper water struggle not to drown and also saw agents pushing migrants into the river on the Del Rio side.

The plaintiffs also said that they lacked enough food and water, which led many migrants to fall ill and, in some cases, to cross the river back to Mexico to find food. The Biden administration said at the time that it was providing food, water and medical treatment to the migrants in a makeshift encampment under and around a bridge.

The situation was a critical moment in Mr. Biden’s first year of handling surging numbers of migrants at the border. The administration’s response, which included the expulsion of thousands of Haitians, galvanized civil rights groups and others to press for better treatment of Black migrants in particular. According to recent government data, nearly 16,000 Haitians crossed into Del Rio in September; about 40 percent of them were expelled under the public health rule, known as Title 42.

The complaint describes squalid conditions under the Del Rio International Bridge, where many spent more than a week when temperatures regularly exceeded 100 degrees.

Jen Psaki, the White House press secretary, said on Sept. 20, “Obviously, any circumstance where individuals are not treated humanely, whether they are coming to our border or not, is not in line with the Biden administration policies.”

Christopher Cameron contributed reporting.