DEMING, N.M.—Chris Brice was relaxing on his couch on a recent Saturday when he got a text that the Border Patrol would be dropping off a large group of migrants in this small desert town the next day.

In less than 24 hours, the local government official helped transform an empty, tin-roof fairgrounds hall on the edge of Deming, population 14,000, into a bare-bones shelter for hundreds of families who crossed the border illegally to request asylum. When Border Patrol agents arrived with two busloads of migrants on Mother’s...