The seven-year-old case reached the Supreme Court earlier this year but the nine judges did not rule on whether the family of the slain boy, Mexican nationals, have the right to sue the US agent in US courts.
Instead, the high court justices sent the thorny issue back to the New Orleans federal appeals court to rule based on established precedent from other court decisions.
Sergio Hernandez, 15, was shot dead by border patrol agent Jesus Mesa on June 7, 2010 as he and three friends played in the dry riverbed of the Rio Grande that separates the Mexican city of Ciudad Juarez from its Texan neighbor El Paso.
It was not the first time: according to the family's lawyers, US border patrol agents have fatally shot at least eight Mexicans between 2006 and 2016 in cross-border incidents.
With the support of the Mexican government and the American Civil Liberties Union, the family sued, based on US federal law offering protection against the disproportionate use of force by law enforcement.
But that law applies to Americans inside the country. Because the victim was a Mexican national, and died in Mexican territory, the US courts have said so far that they have no jurisdiction over the matter.
After two hours of arguments Wednesday, judges in the New Orleans court gave no sign of how they would rule.
The issue is complicated as the unmarked border runs through the middle of the Rio Grande culvert, whose banks are concreted up and topped with fences. The area is maintained and policed by both sides.
Some Supreme Court justices asked in hearings earlier this year whether that means US law could extend several meters into official Mexican territory.
But if the court allows the Mexican family to sue US agents, jurists worry that could open the door to a barrage of lawsuits from abroad against US officials, including soldiers, in US courts.
In June the Supreme Court ruled that former top security officials of the US government could not be sued by post September 11 detainees for abuse and torture in US hands.
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