Up to homotopy principal bundle

Question

A principal bundle (say, in the category of topological spaces and topological groups) induces homeomorphisms on the fibers. This works well for groups $G$. I am wondering if there is an analogue for "principal bundles up to homotopy". Such a structure would only have homotopic fibers. I would also expect this construction to work for higher groupoids. There would be a classifying space $\mathcal{B}(\mathcal{G})$, where $\mathcal{G}$ is some kind of higher groupoid, and where maps $X \rightarrow \mathcal{B} ( \mathcal{G})$ correspond to having $\mathcal{G}$ act on the fibers. (seems related to the suspension functor, perhaps?).