Armed Mexican troops and police regularly stray across the U.S. border, according to statistics the Department of Homeland Security provided to Congress on Tuesday that indicate more than 500 of them have jumped the border in the past decade.

Gil Kerlikowske, commissioner of U.S. Customs and Border Protection, said in a letter to Rep. Duncan Hunter, California Republican, that, in many of those cases, the armed Mexican military or law enforcement personnel ended up in confrontations with American authorities, and 131 people were detained.

The revelations come as U.S. officials are trying to secure the release of Marine Sgt. Andrew Tahmooressi, who has been held by Mexican authorities since the beginning of April, when he drove through an official port of entry into Mexico while carrying firearms.

“While the number of unauthorized incursions by Mexican authorities is relatively few, it is imperative for our officer safety to handle each situation assertively but with sensitivity and professionalism,” Mr. Kerlikowske said.

But Mr. Hunter disputed the commissioner’s evaluation that dozens of incursions a year was rare.

“DHS states that the number of incursions is ‘relatively few,’ but that is a misrepresentation of the frequency of these occurrences, which Mexico invites through its activities along the international border. Also, there is a clear lack of consistency among DHS in handling these incidents, especially in cases of unauthorized incursions with armed authorities,” Mr. Hunter said.

According to Homeland Security numbers, there have been 300 incursions by Mexican police or troops since Jan. 1, 2004. The Mexicans were armed in slightly more than half of those incidents, totaling 525 people. There was a verbal or physical altercation between U.S. authorities and the Mexicans in 81 instances — totaling 320 Mexican police or troops.

In the case of Sgt. Tahmooressi, Mexican officials said he didn’t immediately identify himself as a member of the U.S. military when he drove across the border and into Mexico through an official port of entry.

A petition on the White House website demanding Sgt. Tahmooressi’s release earned nearly 125,000 signatures in May, or more than enough to force a White House response — though when that will come is uncertain.

The debate over incursions comes even as tensions on the U.S. side of the border have escalated amid a surge of children from Central America trying to cross the border, fleeing violence and poverty at home and hoping to reunite with families here.

President Obama and his top lieutenants are trying to figure out a strategy to halt the flow while struggling with how to care for the children who have already reached the U.S.

Vice President Joseph R. Biden will travel to Guatemala later this week to urge parents not to send their children on the treacherous journey, and the administration insists the kids will not be eligible for either Mr. Obama’s nondeportation policy for young adult illegal immigrants nor the path to citizenship contained in the immigration bill that passed the Senate last year.

But some politicians want the administration to be more definitive and ensure the children will be deported quickly.

On Tuesday, former Secretary of State Hillary Rodham Clinton added her voice to those calls for clarity.

“They should be sent back as soon as it can be determined who the responsible adults in their family are because there are concerns about whether all of them can be sent back, but I think all of them that can be should be reunited with their families,” she said at a town hall hosted by CNN.

She said the U.S. must do more to confront violence in Central America and must stiffen border security here but insisted a stern approach to sending the children home will also be needed.

“We have to send a clear message that ‘Just because your child gets across the border, that doesn’t mean your child gets to stay,’” she said. “We don’t want to send a message that is contrary to our laws or encourage more children to make that dangerous journey.”

For its part, the White House was accused of stoking mixed messages when it hosted 10 of the young adult illegal immigrants, who call themselves Dreamers, at an event Tuesday. They are in the U.S. under Mr. Obama’s “deferred action” policy, which has granted more than 500,000 of them amnesty from deportation and has given them work permits.

“These eligible young people are American for all intents and purposes other than the country where they were born,” said Alejandro Mayorkas, deputy secretary at the Department of Homeland Security.

The children pleaded with Mr. Obama, who was not present, to extend his policies so their parents will also be free from danger of deportation or can come back to the U.S. to reunite with their children.

“This year it’s going to be nine years that I have not seen my mom,” said Sarahi Espinoza, who choked up as she recalled her own story of family separation from her mother, who is back in her home country of Mexico.

Critics, though, said giving the children a platform at the White House will lead to more children trying to jump the border.

“How can we expect to dispel rumors throughout Central America that children who enter America illegally will be allowed to stay while simultaneously touting the success stories of a few illegal immigrant children granted de facto amnesty by the administration?” said Rep. Candice S. Miller of Michigan, a senior Republican on the House Homeland Security Committee.

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