“We’re still going, because we don’t think that just because they’ve moved them to a different facility that makes it any better,” said Julie Lythcott-Haims, an author from Palo Alto, Calif., who was driving in her white Jeep Wrangler with a group of newfound friends toward Texas for a protest in Clint that they had planned for Tuesday evening.

She added that at least a dozen people were planning to travel from California and New York. “We want to be there with our bodies and our hearts and our brains and show support for the migrant families and children,” Ms. Lythcott-Haims, 51, said.

The Border Patrol was routing children to Clint because the agency had been facing an unusually large influx of border crossers. The infants there had either been separated from adult family members with whom they had crossed the border or were the children of teenage mothers who had also been detained there. Some of the minors had been held there for nearly a month.

The lawyers who visited the facility, first interviewed by The Associated Press and later by The New York Times, said that children lacked access to private bathrooms, soap, toothbrushes or toothpaste. Many were wearing the same dirty clothes that they had crossed the border in weeks earlier.

Some sick children had been quarantined in Clint, and the lawyers who traveled there were allowed to speak to those children by phone, but not in person.

Ms. Lopez-Sandoval said that only 30 children remain in Clint. The Border Patrol station there was meant to be temporary; children are supposed to be transferred out after 72 hours. But many had been languishing there because the Department of Health and Human Services’s shelters were full.

The strain on that system began to let up last week as a result of two changes: First, the Mexican government launched tougher efforts to limit the number of border crossers from Central America, under pressure from President Trump. Second, the Department of Health and Human Services scaled back a policy requiring fingerprints from family members who applied to sponsor children in its care, speeding up the children’s release from government facilities. Many of those who apply to sponsor such children are undocumented themselves and unwilling to submit identifiers that could put them or their family members in danger of arrest and deportation.