In early October, members of the Family Ties Social Aid and Pleasure Club strutted through downtown New Orleans in fresh new blue outfits. Under the day’s hot sun, sweat poured off foreheads and some suits wilted a little. Still, the club was prepared to keep up their fancy footwork for the standard second-line parade length—four hours. But the New Orleans Police Department shut them down 30 minutes early, they say, with no explanation.

Nearly two hours earlier, the club h...