James McHenry, the director of the federal immigration courts, testified that unaccompanied minors were about twice as likely as other migrants to fail to appear in court. More than half miss their hearings and are ordered deported “in absentia,” though most of those remain in the country anyway without legal status.

The officials who were questioned argued that migrant children become the responsibility of local child welfare agencies when they are released to sponsors. But the senators pushed back, pointing out that the federal government does not alert those authorities when migrant children are placed in their jurisdictions.

“You want to talk about catch and release?” said Senator Claire McCaskill, Democrat of Missouri, referring to a term used by Mr. Trump and other critics of the Obama administration’s policy of releasing migrant families while they await asylum decisions. “You’re catching these children and then you’re releasing them, and everyone goes like this: ‘Not my problem.’”

Children traveling alone surged across the southern border under Mr. Obama in 2014, when border agents apprehended 68,541 trying to enter the United States. About 41,000 have been apprehended in the current fiscal year, a 16 percent increase from the same period in 2017.

Since the congressional investigation began in 2015, the Office of Refugee Resettlement, part of the Health and Human Services Department, has strengthened the background checks that it conducts on sponsors before releasing children to them. The agency also began the practice of placing a phone call to sponsors one month after a child’s release.

Commander White, the department official, acknowledged that follow-up ends there, and most of the phone calls go unanswered. He said that sponsors are not required to stay in contact with the agency, and many “do not want anything more to do with the systems that they have been through.”

Senator James Lankford, Republican of Oklahoma, asked whether fewer children might travel to the United States on their own if the government required sponsors to be in the country legally. Commander White declined to speculate on whether such a rule could deter migration over all, which is a well-known goal of the Trump administration. In the short term, he said, it would create a “backup into border stations and produce a humanitarian crisis.”