Railing against President Biden’s border policies, Mr. Trump added once again that many children were now “dead” and “if it were me, it would be the biggest story.”

In fact, when Mr. Trump was in office, his administration also tried to contact — and was also unable to reach — thousands of migrant children who had left the refugee office’s care. And there is no evidence that thousands of the children died under either Mr. Biden or Mr. Trump.

Under Mr. Trump, the Health and Human Services Department could not reach 19 percent of migrant children (about 1,500) who were placed with sponsors from October to December 2017, or 11.6 percent of children or their sponsors (about 9,200 calls) from August 2018 to December 2020. In comparison, a Biden administration official testified to Congress that 19 percent of the refugee office’s calls to the children or their sponsors went unanswered.

When Democrats accused the Trump administration of “losing” unaccompanied children, the health department said in 2018 that those claims were “completely false” and that sponsors “simply did not respond or could not be reached when this voluntary call was made.”

At the time, a Trump administration official said that there were various reasons sponsors or children might not pick up the phone. They may be hesitant to answer calls from unknown government agents, especially if they have unauthorized status or a fear of traffickers. Sponsors may also simply have different telephone numbers.

“We have 88,000 missing children. Now, can you imagine if that were Trump that had 88,000 missing children? 88,000. That’s a holocaust.”

“The Biden-Harris administration has lost track of an estimated 150,000 children, many of whom have undoubtedly been raped, trafficked, killed or horribly abused.”

Mr. Trump appeared to extrapolate this figure from government data showing that nearly 400,000 children had been placed with sponsors over the past four years. Assuming that the office was still unable to reach a third of them, that is equivalent to about 133,000 children.

A report from the Department of Homeland Security’s inspector general provided Mr. Trump with new fodder. The report, released in mid-August, noted that out of some 448,000 unaccompanied migrant children placed into the custody of the refugee office from the 2019 to the 2023 fiscal years, about 32,000 children had failed to appear for their immigration hearings. Another 291,000 did not receive “notices to appear” in immigration court at all.

Again, these children were not “lost,” though the report chided immigration officials by noting that without an ability to monitor the location of the children, they were more susceptible to “trafficking, exploitation or forced labor.” Moreover, about half of that period — the 2019 and 2020 fiscal years, and part of the 2021 fiscal year — was during the Trump administration.

“According to a new D.H.S. report, Kamala Harris also lost, and this is impossible to believe, listen to this, 325,000 migrant children are gone. They’re missing. She allowed them to be trafficked into our country.”

By now, the claim, updated with the inflated figure, has become a common refrain for Mr. Trump.

“Under this administration, 325,000 migrant children are missing, 325,000. In other words, take your biggest stadium in California and you could fill it up five or six times. Those are all missing children.”

“This is not even possible to believe. 325,000 migrant children are missing, many of whom have been trafficked and raped. As California attorney general, she lost them all.”

Mr. Vance, Mr. Trump’s vice-presidential pick, advances the talking point on the national stage, though he uses more tempered language in claiming that the children have been “effectively lost.” He does not repeat Mr. Trump’s contention that “many” have died.

“She doesn’t mention how bad inflation was, how bad the 325,000 children that we just mention.”

Mr. Trump abbreviates the misleading claim into a simple two-word phrase: “325,000 children.” Like other phrases — “laptop from hell” or “$85 billion in equipment” — the shorthand suggests that his audience is now familiar with his distortions after months of repetition.