Suppose that $$F:\mathcal{D} \to \mathcal{C}$$ is a Grothendieck fibration of small categories, whose fibers are groupoids. Is there anything sensible which can be said about the induced morphism $$BF:B \mathcal{D} \to B\mathcal{C}$$ between classifying spaces? It seems that Quillen's theorems do not buy us much when the fibers are groupoids, since fibers are homotopy equivalent if and only if they are equivalent as groupoids. I also care about the special case of discrete fibrations (i.e. the Grothendieck construction of a presheaf of sets). Is there some way of expressing the homotopy type of $B \mathcal{D}$ in terms of the homotopy type of $B\mathcal{C}$ together with the data of the psuedofunctor $$\varphi:\mathcal{C}^{op} \to \mathcal{Gpd}$$ classified by the fibration $F,$ which is computationally tractable, at least for "nice enough" $\varphi$? I am aware of a spectral sequence involving the homology of a $\mathcal{C}$module derived from $\varphi$ converging to the homology of $B\mathcal{D},$ but I would really like something computable from knowing only the homotopy type of $B\mathcal{C},$ not $\mathcal{C}$ itself.

Since others seem to be interested in this question, p. 19 of this preprint http://arxiv.org/abs/1112.3996 appears to have nice spectral sequences for homology (resp.cohomology) of $B\mathcal{D}$ in terms of homology (resp. cohomology) of $B\mathcal{C}$ with coefficients twisted by a local system determined by $\varphi.$ Actually, I am a bit confused. Could anyone tell me why $H^{q}\left(BD_\left(\mspace{3mu} \cdot \mspace{3mu}\right),A\right)$ is a local system? It seems like I would need $\varphi$ to invert all morphisms. 

