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 $\varphi.$

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.