Okay, so in continuing to figure out how these rules interact, I re-read the section on Stealth.
You make a Stealth check to hide, which costs an action.
But you also make a Stealth check to sneak past some guards, which can be done on your move.
So it strikes me as reasonable to say that a creature who has just gone invisible should be able to make a Stealth check to attempt to move somewhere quietly so that others in the area don't know where it ends up. The DC would be either the highest Passive Perception of the others (if they're not actively tracking it) or the highest Perception roll of anyone who is tracking it.
Agree/disagree?
This is, of course, a consequence of folding both move silently and hide into one skill. And I'm okay with the folding, just trying to figure out how the new version works best.
If you take the hide action while invisible, what are you actually doing in-game, and how does it differ from a normal stealth check?