What limits do commute with pushouts in Set?

I know of the following two colimit/limit commutation results in the category of sets: Products commute with sifted colimits and finite limits commute with filtered colimits.

Does someone know classes of limits that commute with pushouts in Set?

Background: I have a certain class of objects in a category and want to know whether it is closed under pushouts. I seem to be able to define this class as those objects that are limits of certain special objects, analogous e.g. to profinite spaces being limits of finite discrete ones. My question is whether such classes are closed under pushout, then I could construct a lot of new examples.

Solution 1:

This MO question discusses the commutativity of pushouts and pullbacks and might be helpful to you (for Trimble's answer, note that the category of sets forms a "topos"): https://mathoverflow.net/questions/46230/distributivity-commutativity-of-pushouts-and-pullbacks