Sure.
I'm not sure all mods would necessarily check out external links. Some have policies against I think. Also we actually only actually moderate what goes on here on this board, but often additional context can be found through external means.
So, while no action will be taken regarding things that occur on another site, if there is context or a strong enough relation/link to stuff that's going on on this site, we'll act on it certainly.
Good examples might include - like if somebody's planning a spam attack on this board using another forum and we catch wind of it, we can often head that off. Or somebody posting information about things they did on this board that pan out. Or harrassment that seems to be occurring.
Again, in that case, we can't act directly on stuff that happens on other boards. But if it's clear that the additional evidence supports the notion that it's going on here, then that will be taken into account in taking action against somebody for harrassment potentially.