Let $F : \mathcal{D} \to \mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $\mathrm{colim}_\mathcal{D} F$ pulls back to a local system $M_d$ on $F(d)$ for each $d \in \mathcal{D}$, and also a local system $M_h$ on $\mathrm{hocolim}_\mathcal{D} F$.

Is there a Bousfield-Kan type spectral sequence of the form $$E^2_{s,t} = \mathrm{colim}^s_{\mathcal{D}} H_t(F(d);M_d) \Rightarrow H_{s+t}(\mathrm{hocolim}_\mathcal{D} F;M_h)$$ and if so where can one find it in the literature? I would also be content to know if this is not possible.