WASHINGTON—The Trump administration requested $4.5 billion from Congress to respond to the growing surge of migrants at the nation’s southern border, deepening a fight with Democrats over a crisis that both sides accuse the other of exacerbating.

The thousands of asylum-seeking Central American families arriving each day at the border, fleeing poverty and violence, have brought the border infrastructure to its breaking point, according to top officials, including acting Homeland Security Secretary Kevin McAleenan.

The administration said Wednesday that a 16-year-old Guatemalan boy died in U.S. government custody this week, the third immigrant child to die in custody since December. The boy, Juan de Leon Gutierrez, died in a Texas hospital Tuesday, following several days in the intensive care unit, another government official said. The boy had crossed the border illegally near El Paso, Texas, about two weeks earlier and had been held by U.S. Border Patrol for several days before being transferred to a government shelter, the official said. The cause of death and the child’s time in the government’s care are both under review by authorities.

Mr. McAleenan, previously the commissioner of Customs and Border Protection, had warned that without changes and more resources, it would be only a matter of time before another child died in custody soon after arrival. Facing dwindling options for how to handle newly arriving migrants, who often cross between ports of entry in large groups with children, the administration is also seeking to discourage such travel by making it harder for them to seek asylum.

The administration said Wednesday it would begin using voluntary DNA testing to establish that children traveling with adults were related to them at certain locations on the southwest border, based on suspicion that some adults were bringing children to whom they weren’t related.