Born in Great Lakes, Branch was raised in Los Angeles, California, but in 1969 he moved to Chicago where he attended the University of Illinois. He soon took the place of the harmonica player Carey Bell in a band led by Willie Dixon called the Chicago Blues All-Stars.

In the 1970s he founded his own group, The Sons of the Blues, along with Lurrie Bell on guitar and Freddie Dixon on bass guitar. They are the sons of Carey Bell and Willie Dixon respectively, and they recorded for Alligator Records and under new personnel for Red Beans Records. The new band consisted of Carlos Johnson on guitar and J.W. Williams on vocals and bass guitar. He has also recorded for Verve Records and Evidence Records.

Other than co-headlining Alligator's 1990 summit meeting Harp Attack! with fellow harp masters Junior Wells, Carey Bell, and James Cotton, Branch largely busied himself with extensive sideman work (he's first-call session harpist around the Windy City) and teaching an innovative "Blues in the Schools" program until 1995.

Branch has appeared at numerous major festivals including the Long Beach Blues Festival, Chicago Blues Festival, and the San Francisco Blues Festival.