So I am currently trying to piece together little bits of knowledge I have acquired about fibrations in various context when I came across this question. If $p : E \rightarrow B$ is a Serre fibration and $f \simeq f' : X \rightarrow B $ are two homotopic maps, is $X \times_f E$ (weakly) homotopy equivalent to $ X \times_{f'} E $?
I know that if $p : EG \rightarrow BG $ is the universal $G$ bundle than the pullbacks are isomorphic as bundles. There is no way anything this strong can be true in general though.
If any of this is true how much of this holds in the setting of model categories? Do they have to be right proper?
Any bits of info would be greatly appreciated
Yes, a pullback square with one map a fibration has the property that a homotopy pullback has the same homotopy type as the actual pullback. Namely, changing the homotopy type of the bottom map does not affect the pullback.
See https://ncatlab.org/nlab/show/homotopy+pullback Proposition 3.1 for details on when this holds.