Supreme Court justices will grapple Tuesday with how far American constitutional protections extend outside the U.S. as they hear oral arguments on a case that saw a Border Patrol agent fire his gun into Mexico, which struck and killed a 15-year-old boy.

Agent Jesus Mesa Jr. said the boy and his friends were throwing rocks as they tried to sneak into the U.S. in 2010.

The family of Sergio Adrian Hernandez Guereca said the teens were only playing a game where they would run up to touch the fence on the U.S. side of the border in El Paso, Texas, before running back into Juarez, Mexico.

Now the family wants permission to sue Agent Mesa for damages, saying he violated their son’s Fourth and Fifth Amendment rights.

Should the family prevail, it could spark more uncertainty in how far constitutional rights extend.

“I think it could be a very significant ruling, but again, it would just mean more lawsuits, more legal uncertainty,” said Andrew Kent, a professor at Fordham University’s School of Law.

The case comes at a time when federal courts are already grappling with who has constitutional protections in light of President Trump’s extreme vetting executive order.

Several courts have ruled that immigrants, both legal and illegal, and even visitors who came to the U.S. but have since left, can claim U.S. constitutional rights — a finding that immigration crackdown advocates say breaks with decades of precedent.

In the border case, Agent Mesa was cleared by the Justice Department for the shooting. That probe found that “smugglers attempting an illegal border crossing hurled rocks from close range at [Agent Mesa,] who was attempting to detain a suspect.”

The department said Agent Mesa didn’t break Border Patrol policies on use of force. The agent is still on the job today.

Mexico indicted Agent Mesa on murder charges, but the U.S. refuses to extradite him.

The Hernandez family argued that Mr. Mesa violated their son’s Fourth and Fifth Amendment rights under the U.S. Constitution by killing him, but the 5th U.S. Circuit Court of Appeals ruled that those rights don’t extend to noncitizens outside the U.S.

The family says a 2008 case involving war on terrorism suspects being held at the Guantanamo Bay prison established that even noncitizens have constitutional rights, and violations can be considered, based on the totality of the circumstances.

Agent Mesa’s lawyers, though, point to a 1990 case, which ruled those outside the U.S. don’t have constitutional protections against searches and seizures.

Mr. Kent, who has been observing the case, said the decision may come down to Justice Anthony M. Kennedy’s swing vote.

“Justice Kennedy is a pretty strong believer that [with] the Constitution, there shouldn’t be a bright line rule that it shouldn’t apply in any given place,” said Mr. Kent.

Justice Kennedy argued for that point in the 2008 Boumediene v. Bush case that granted habeas rights to prisoners at Guantanamo.

“He believes that pretty strongly, and I think the four liberal justices will support him on that,” said Mr. Kent.

Mr. Kent said the Hernandez family could win its claim that the boy’s rights were violated, but still lose their demand that the federal government be responsible for monetary damages.

Agent Mesa’s lawyers said in their briefs that the 15-year-old boy had been twice arrested by U.S. border authorities for alien smuggling, but both times had been released back into Mexico because of his juvenile status.

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