TUCSON, Arizona — 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 in June, the two of them set out from Guatemala anyway.

The truth, he said, moments after they arrived at a 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 Mr. Cruz, and has played a major role in driving thousands of families across the border with Mexico.

The policies of President Donald J. Trump — first an effort to separate families crossing the border, and now an effort to loosen restrictions on detaining migrant families — represent an aggressive effort to rescind that invitation. (More recently, the government said it would temporarily stop turning families over for prosecution, but the White House said its immigration stance was unchanged.)

But interviews along both sides of the border, as well as a review of immigration numbers, suggest that it’s going to be difficult for the president to stanch the flow.

One thing was clear at the Arizona-Mexico border: Many families — especially those from countries in Central America plagued by gang violence and ruined economies — are making the calculation that even separation or detention in the United States is better than the situation at home.

«Why would you undertake such a dangerous journey?» said Magdalena Escobedo, 32, who works at the migrant shelter here in Tucson, called Casa Alitas. «When you’ve got a gun to your head, people threatening to rape your daughter, extort your business, force your son to work for the cartels. What would you do?»

More than 250,000 migrants had been arrested this year as of late May, according to government data; that is close to the total number arrested in all of 2017, about 311,000.

Casa Alitas takes in families who’ve presented themselves to border officials to ask for asylum. Once they’re processed, authorities drop them off here for a meal and a shower before they head off to stay with friends or relatives and wait for their day in court.

Men like René Pérez, 40, who made it into the United States with his son, said he was well aware that they might have been separated or placed in custody. «If it happens, it happens,» Mr. Pérez said.

Across the border in the Mexican town of Nogales, many parents said temporary separation from their children in America would be better than facing the violence back home.

«I’d rather accept that, to know that my son is safe,» said Lisbeth de la Rosa, 24, who was waiting to enter the United States with her 4-year-old son.

«It’s worth it,» said Lidia Rodríguez-Barrientos, 36, with her 9-year-old daughter. «Because we’re afraid to go back.»

Critics have long said that allowing migrants to go free while their immigration cases are pending encourages parents to enter the United States with children, and some conversations bear that out.

«This is the reason I brought a minor with me,» said Guillermo T., 57, a construction worker who recently arrived in Arizona. Facing unemployment at home in Guatemala, he headed north; he had been told that bringing his 16-year-old daughter would assure passage. He asked that only his first name be used. «She was my passport,» he said of his daughter.

On June 21, federal officials dropped Mr. Cruz and his 16-year-old son, also named Luis, at Casa Alitas. The elder Mr. Cruz, a lemon and orange grove worker, had hoped to live his life in his home state of Suchitepéquez. But in May, his son was approached by a gang who demanded he join them, flashing a gun. «They kill you if you don’t obey,» Mr. Cruz said.

On June 3, the pair left for the United States, and presented themselves at the border to ask for asylum. The younger Mr. Cruz revealed a letter his school director had written before he left — a note they hoped would be the evidence that they needed asylum in America.

«The student had to withdraw himself from school due to violence and gang persecution,» she wrote. «He decided to move to save his life.»

‘If they separate us, they separate us,’ one parent says.

© 2018 The New York Times