If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow holim_IX\circ F$ is a weak equivalence if all the categories $F/j$ are contractible.
For $C=Top_\ast$ the category of (compactly generated Hausdorff) based spaces, is there a theorem about the connectivity of $holim_JX\longrightarrow holim_IX\circ F$? Thinking of holim as a hom object, the connectivity should depend on a notion of "connectivity for the diagram X", and of "relative dimension" for the map of $J$-shaped diagrams of spaces $BF/(-)\longrightarrow BJ/(-)$.
As a baby example, let $\mathcal{P}(n)$ be the poset of subsets of the set with $n+1$ elements, and $F\colon \mathcal{P}(n)\backslash\emptyset \longrightarrow \mathcal{P}(m)\backslash\emptyset$, for $n\leq m$, the inclusion that sends $S\subset n$ to $S\cup(m\backslash n)$. Let $X\colon \mathcal{P}(m)\backslash\emptyset\longrightarrow Top_\ast$ be the diagram with $X_m=Y$ for some fixed based space $Y$, and $X_S=\ast$ a point for all proper subsets $S\subset m$. Then the canonical map on homotopy limits is the restriction map $\Omega^{m}Y\longrightarrow \Omega^nY$ between loop spaces. This is as connected as the connectivity of $Y$ minus the dimension of $S^m/S^n$.
