Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to each stratum is a fibration. Do the actual fibers of $f$ compute the homotopy fibers?
I think so. Let's look at the first case. Suppose that $X$ has a closed subspace $A$ such that $X$ is the homotopy pushout of $X-A\leftarrow N\to A$ for some bundle $N$ over $A$. If both $X-A\to Y$ and $A\to Y$ are fibrations, then so is $N\to Y$. Now since $X-A\leftarrow N\to A$ is a diagram of spaces mapped compatibly to $Y$ by (quasi)fibrations it follows that $hocolim(X-A\leftarrow N\to A)\to Y$ is also a quasifibration.