It's not a stretch to say that music—and especially drumming—is in Joe Lastie’s genes.

As a child, he heard his maternal grandfather playing drums in church and his paternal grandfather, Frank Lastie, do the same. Other drummers in his family include greats such as his uncle Walter “Popee” Lastie and his cousin Herlin Riley. He also boasts kinship to trumpeter Melvin Lastie, saxophonist David Lastie, R&B man Jessie...