As a result, the vast majority of migrants who illegally enter the United States are told to report to court at a later date and then are released. From there, the migrants search out relatives or other places to stay while their cases wind their way through immigration court.
Last month, the Trump administration began flying migrants from overcrowded centers elsewhere in the Rio Grande Valley in Texas to Del Rio to be processed. But agents in Del Rio said the processing facility there was also over capacity; this week, the authorities there built a tent city to house migrants.
Storage areas that had been used for Border Patrol equipment in Del Rio “are now storing baby formula, diapers, additional extra space for blankets, heat blankets they can wrap up in,” Chief Davis said last month. That day, agents across the sector arrested more than 300 migrants.
Most of the migrants who are caught by Border Patrol come to the United States to escape poverty in their home nations, Mr. Kemmett said, and do not appear to be the criminals that Mr. Trump has warned of in his campaign to limit immigration.
Agents in Del Rio said there was no sign that the surge of migrant families would decline.
Mr. Kemmett said the Border Patrol agents he oversaw were becoming “battle fatigued.” He also questioned why migrants “would knowingly put your child in that type of danger, with the risk of losing your child,” in attempting the perilous journey to the United States.
For some of the immigrants, it is the only way to protect their families.
Barefoot and balancing his 8-year-old daughter on his shoulders, Ángel Gabriel, 39, crossed the Rio Grande and into Eagle Pass on a hot afternoon last month. He steadied his 32-year-old wife as they walked into the river current; she tightly gripped their barefoot 16-year-old son.
Border Patrol agents met the family in the middle of the river and brought them to land. As soon as agents finished patting down Mr. Gabriel to ensure he was not carrying weapons or drugs, he rushed to his family and embraced them.