While unraveling the formalism of descent in basic settings (see this question), I came across two different kinds of isomorphisms which involve pullbacks and coproducts.
One is $$X\times_U\coprod_iU_i\cong \coprod_i(X\times_UU_i),$$ and the other is $$\coprod_iU_i\times_U\coprod_iU_i\cong \coprod_{ij}U_i\times_UU_j.$$
Intuitively, I'd expect commutation of pullback with coproduct to be something like $$\coprod_iA_i\times_{\coprod_iC_i}\coprod_iB_i\cong \coprod_{i,j,k} A_i \times_{C_j}B_k,$$ but that doesn't seem quite right either.
What is the proper way to state commutation of pullbacks with pushouts, and does the second "equation" hold in the category of topological spaces (the first one does, see [2])? If so, why?
The point is that Top is Cartesian closed (assuming you're using the right Top.) Thus pullback functors have right adjoints and preserve colimits. Since the forgetful functors from the slice categories to Top also have right adjoints, they preserve colimits too, so for instance your first line follows. For the second fact, use this commutativity twice; get $\coprod(U_i\times_U \coprod U_i)$ and then the coproduct of all pairwise pullbacks, as desired. For the third, it's enough to calculate $A\times_{\sqcup C_i} B=\sqcup A\times_{C_i} B$, using that coproducts in Top are disjoint.