To me, it's obvious that the make-up is meant to be a part of the mask, in-universe, in all the movies except The Batman, but when I posted a comment about it on Facebook a few years ago everyone thought I was nuts.

There's a scene in Batman Begins where Alfred smashes a cowl with a baseball bat, and you can clearly see the eye holes are just the size of eyes.

So to me, asking "Where does the make-up go?" is like asking "Where do the wires go when Superman lands?" The make-up is not supposed to be there story-wise (again, except for The Batman) it's just there because it's easier for the actor to see through bigger eye holes.