McALLEN, Texas—They come in pairs, worn-out migrants carrying—or, in some cases being led by—children who range in age from infants to teenagers. Many of the adults weep openly when the doors to the parish hall of the Sacred Heart Catholic Church in McAllen swing open and nuns, Jesuit priests and a host of local volunteers rise to their feet in a raucous standing ovation.

"Bienvenidos!" shouts volunteer Hermi Forshage, clapping. "Bienvenidos! Welcome!"