In compactly generated weak hausdorf spaces (CGWH), proclusions are compatible with base change. (A proclusion $f:X\to Y$ is a map of topological spaces which is homeomorphic to a quotient map.) That is, if $f:X\to Y$ is a proclusion between CGWH-spaces, and $B\to Y$ any map between CGWH-spaces, then $A=X\times_Y B\to B$ is also a proclusion, where the fiber product is taken in CGWH. (Note that $A$ is set theoretically the fiber product, with the topology obtained by replacing the product topology with the closest compactly generated one.)
So in CGWH, quotients are always compatible with base change.
In Top, a colimit is always be computed as a quotient of a coproduct. Unfortunately, this is not true in CGWH; the colimit in Top of a diagram of CGWH-spaces will be compactly generated, but not necessarily weak hausdorf.
There are some results which will tell you that a certain colimit in Top of CGWH-spaces is also the CGWH-colimit. For instance, given $f:X\to Y$ and $g:X\to Z$ maps of CGWH-spaces, if $f$ is the inclusion of a closed subspace, then the Top-pushout of $f$ along $g$ is actually in CGWH. Since the pushout is a quotient of the coproduct of $Y$ and $Z$, you can see that this particular pushout is compatible with fiber products, by the proclusion result I mentioned above.
If you're looking for a counterexample in CGWH, I'd look for a Top-pushout of CGWH-spaces which is not weak hausdorf. I don't have a particular one handy, though I'm sure there are many.
Most or all of the above comes from Gaunce Lewis's thesis, though I don't have it at hand to give precise references.