WASHINGTON — Attorney General William P. Barr moved on Monday to end asylum protections for migrants solely because their relatives have been persecuted, the latest attempt by the Trump administration to limit sanctuary for people seeking refuge in the United States.

Mr. Barr’s decision overturned a 2018 judgment by the Justice Department’s Board of Immigration Appeals, which found that a Mexican migrant whose father was targeted by a drug cartel could be eligible for asylum.

Migrants are eligible for asylum in the United States if they can prove they were persecuted because of their race, religion, nationality or what immigration laws describe as “membership in a particular social group or political opinion.” Mr. Barr’s ruling concluded that the immigration appeals court “erred” in finding that a migrant’s family qualified as a persecuted social group.

“I conclude that an alien’s family-based group will not constitute a particular social group unless it has been shown to be socially distinct in the eyes of its society, not just those of its alleged persecutor,” Mr. Barr concluded.