“I wanted to be the main taco,” Dwayne Dopsie says with a laugh, of his very early desire to not only play the accordion but also lead his own band like his father, the late Alton “Rockin’ Dopsie” Rubin. “I always loved how people gave him so much attention—watching him playing, watching his fingers. So I wanted to follow in his footsteps.”

Whenever Dwa...