So I don't think the USAF heritage flight is much like the BBMF, but happy to be correced.
Sad to say you are correct Sandie. The only part of the USAF Heritage Flight that is operated by the US Air Force are the current line fighters, all the other aircraft are privately owned or civilian owned interests.
Shame really, as I would loved it if the USAF had saved a couple of B-36s and B-47s and kept them in flying condition.
By the way, there are civilian owned F-100s, F-4s and F-104s flying in the US, along with F-86s and I do believe one F-84.