A general homological Kleiman-Bertini theorem

Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g is an element of G, the sheaf Tor(j)(X)(g F, E) vanishes for all j >= 1. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman-Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.