Those bands had revolving door memberships (hence being named after leaders) and some were not even actual bands, but names made up for specific recording sessions. For example, Jackie Brenston was a frequent sax player for Ike Turner's bands and the "band" Jackie Brenston and His Delta Cats existed for only one recording session, the famous
Rocket 88 single. It was really Ike Turner's sidemen, but renamed for contractual reasons.
Little Richard's Band featured different musicians in almost every tour.
There was no "John Paul George and Ringo" dynamic in any of Hess combos.
A lot of bands have revolving door memberships. Even the Beatles had changes in personnel.
Booker T & the MG's was a "black" band--although I'm not sure all of its members were black--that as far as I know had only one member leave from 1962, when it formed until it broke up in 1973.