Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. I would like to know when this induces a homotopy equivalence $$B(\text{colim}\, D)\stackrel{\sim}{\rightarrow}\text{hocolim}\, BD$$ Or more generally when there are known methods of computing the homotopy type of $B(\text{colim}\, D)$ from $BD$.

In the example I have in mind, $I$ is the poset of natural numbers, every space in sight is compactly generated and every functor $D(n)\rightarrow D(m)$ is a cofibration on both objects and morphisms. Futhermore, the indentity map in each $D(n)$ is a cofibration and the source and target maps are fibrations. In particular, in this case the above homotopy colimit is the ordinary colimit. However, I think the more general question is also of interest.

References to the literature are also welcome!