So the dotted lines mean "pupil range" - meaning the bounds of where a pupil can move inside. Usually, if you name something Left Eyeball or Right Eyeball and it's in your eye group, that will get auto-tagged as this range (because you don't usually want the pupil to leave the eyeball).
For your case, I would suggest making a bottom half-circle, essentially the white below the lids, and calling that Left Eyeball or Right Eyeball (or tagging it with the dotted lines). Then the pupil should stay within there. You can fine tune the amount the pupils move with Face > Eye Gaze Strength (or arm mouse input instead to move them with mouse drags).
I'm having a similar problem, keeping a large pupil within the bounds of the eyeball.
Ideally, I'd like the pupil to crop to the shape of the eyeball when it reaches the edges.
Currently trying to manipulate the R and L pupil range to accomplish this, but cant figure out how to grab and manipulate those handles.
But if I do figure that out, it's simply a range right? It doesn't crop the pupil when it gets to the edge?
Glad you figured it out. Pupil range is just one of those things where every character is going to be a little different, so I just play around with the artwork, tagging, and eye gaze settings until finding something that works.