Let $G$ be a (nice enough) topological group (actually a filtered colimit of compact Lie groups), and let $X$ be a manifold with an action (a proper one in fact) by a Lie group $H$. Let $X//H := (X\times EH)/H$ be the Borel construction.

Does the following claim appear in the literature:

Claim: The set of isomorphism classes of $H$-equivariant principal $G$-bundles is given by $[X//H,BG]$

For the case when $H$ is a compact Lie group and $G$ is an abelian Lie group, then this is proved in a 1983 paper by Lashof, May and Segal,

Equivariant bundles with Abelian structure group, Contemporary Math 19 (1983) 167--176.

And some related cases are treated by May in

Some remarks on equivariant bundles and classifying spaces, Asterisque 191 (1990) 239--263 (pdf)

The claim seems intuitively true, but I can't seem to be able to find it, so my intuition may be off. If it is true, then surely May would have mentioned it (perhaps he does and for some reason I missed it).

EDIT: May also treats the topic in chapter VII of the book Equivariant homotopy and cohomology theory, but it seems a compilation of older work.