“We’re just here to support the Border Patrol and show the public the reality of the border,” said Mr. Benvie, 43, who recently came to New Mexico from Minnesota. He said the organization plans to remain on the border until the extended wall proposed by President Trump is built or Congress changes immigration laws to make it harder for migrants to request asylum.

Militias have recently stepped up their activities in New Mexico and other states as the authorities scramble to respond to a surge in families from Central America, with total apprehensions on the border reaching more than 92,000 in March. Elsewhere on the border, the mayor of Yuma, Ariz., declared an emergency this week as the city sought federal and state assistance to deal with migrant arrivals.

Mr. Benvie, the spokesman for the United Constitutional Patriots, declined to specify how many of its members were in Sunland Park, a city in New Mexico about nine miles west of El Paso. He said the group included people with military or law enforcement experience.

“If these people follow our verbal commands, we hold them until Border Patrol comes,” Mr. Benvie said. “Border Patrol has never asked us to stand down.”

The governor of New Mexico, Michelle Lujan Grisham, said in a statement that it was “completely unacceptable” that migrant families “might be menaced or threatened in any way, shape or form when they arrive at our border.”

“It should go without saying that regular citizens have no authority to arrest or detain anyone,” she added.

The video of this week's episode in the New Mexico desert shows Border Patrol agents arriving on the scene at some point after members of the militia have already come into contact with the migrants. Before the arrival of the federal agents, a woman narrating the video tells a man who appears to be a militia member “Don't aim the gun” in the direction of the families. The migrants can be seen kneeling on the ground and embracing one another.