TUCSON, Ariz. — When Luis Cruz left behind his wife, four of their children and the house he’d built himself, he’d heard that American officials might split him from his son, the one child he took with him. But earlier this month, the two of them set out from Guatemala anyway.

The truth, he said this week, moments after they arrived at a cream-colored migrant shelter in Tucson, was that he would rather be apart from his child than face what they had left behind. “If they separate us, they separate us,” said Mr. Cruz, 41. “But return to Guatemala? This is something my son cannot do.”

For years, children and parents caught crossing the nation’s southern border have been released into the United States while their immigration cases were processed, the result of a hard-fought legal settlement designed to keep children from spending long months in federal detention. In the eyes of the Trump administration, this practice has served as an open invitation for people like Luis Cruz, and has played a major role in driving thousands of families across the border with Mexico.

Mr. Trump’s newest immigration policies — first an effort to separate families crossing the border, and now an effort to change the legal settlement on migrant family detention — represent an aggressive effort to rescind that invitation, one that has plunged the nation into a debate about the limits of its generosity.