PARIS—When armed Chechens descended on the provincial city of Dijon this month and occupied a predominantly North African neighborhood, the French police broke up fighting with locals when they could and called in reinforcements—but kept their guns in their holsters.

Over several days, the armed men—seeking revenge for a Chechen teenager allegedly beaten in the area—fired their guns as they assaulted people and damaged businesses. When the violence subsided, police began to detain suspects, including Dijon residents and Chechens…

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