Puppet's eyes are rotating and moving separate from each other

Woah! Freaky! I would show that one with the Adobe folks. Yes, the default blink had a zero so it was blinking super fast, but after disabling it I can see the pupils are rotating as it blinks. Never seen anything like that before!

Note: If you delete "Left Eyelid Bottom" it behaves better in the left eye. No idea why. I am wondering if the triggers and swapsets are playing up in some way. I am not used to the "Blink" layer being under "Lids" (Blink replaces its siblings, so it will hide the lids and not the pupil.. but Blink is also a trigger in a swapset (see "Eyes" swapset) with lots of other layers). So Blink hides its siblings, and then the swap sets also kick in ... and I think it freaks out a bit. My guess is the pupil range goes weird and the pupil tries to cope, causing it to rotate.

I am not sure why your swapsets are so complicated. It is not necessarily wrong, but I don't understand your goal. I suspect it is something about eyelids, swapsets, etc. making it go "strange". But ... yeah ... that is certainly interesting! Oh, I noticed some layers with 1, 2 etc as children without a Cycle Layers behavior added - not sure if on purpose.

No idea how to solve it and keep all your other stuff in place. Here is a blog I did on eyes a while back in case any use. Your one looks much more sophisticated though. https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/

Probably because you have the automatic blink behavior enabled.
Remove it or click on the eyeball to turn it off.

Give it a try.

Thanks.

https://www.dropbox.com/s/2lur6cg00pn7qf4/edit2_Stella%20original%20%21.puppet?dl=0

Here's a screenshot of the issue happening.