“Yeah, this is my hometown,” declares the remarkably talented Cuban-born percussionist and composer Alexey Marti, who has now lived in New Orleans for 14 years. He will be making his first appearance leading his own band at the French Quarter Festival.

“Most people who move to a different country, they're moving many times,” Marti says. “For me, after eight years I decided to make the city my home base. Musically I star...