I consider the following two situations:

Let $B$ be a simply connected space, and $F\to E\to B$, $F'\to E'\to B$ two fibrations with a map $f:E'\to E$ sending fibers to fibers and inducing the identity on $B$ (in my case $f$ is injective but I don't think it matters).

Let $f:B\to B'$ be a map between simply connected spaces, $F\to E\to B$ a fibration, and $F\to E'\to B'$ the pullback with respect to $f$. Here as well, I deal with $f$ being injective.

**Question**: is there some relative version of the Serre spectral sequence that allows me to compute the relative cohomology groups of $(E,E')$ from the cohomology of $(F,F')$ and $B$ (case 1), resp. the cohomology of $F$ and $(B,B')$ (case 2)?

Some reference would be perfect. If some topological assumption on the spaces or maps are required, I am dealing with classifying spaces and Borel constructions applied to smooth actions on manifolds, so it is very well behaved stuff.