Gestures of sympathy, like a trip to the border by Glenn Beck, the conservative radio and television personality who has raised more than $2 million to buy teddy bears, shoes and food for migrant children, were met with scorn and derision. Some anti-immigrant activists responded to news that the government was buying new clothing for the detainees by organizing a campaign to mail them dirty underwear.

“We can’t elect another Republican president in 2016 who gets 27 percent of the Hispanic vote,” said Senator John Cornyn of Texas, the No. 2 Republican in the Senate, referring to the percentage Mitt Romney won in 2012.

Mr. Cornyn voted against the broad immigration overhaul last year but introduced a compromise measure this week with a Texas Democrat from the House, Representative Henry Cuellar, that would speed the deportation of some children while allowing those who request asylum to stay as they await a hearing.