If the eyes distort in any way, check if they are independant. (in rig mode, the two eyes should have a crown next to the layers names).
Or share a screenshot of your rigging.
I modified the original in Photoshop and the problem disappear.
I learn a lot from tutorials on the web.
By the way, thanks for your answers!
Yes, more screenshots or a little video would be helpful. I notice the two eyeballs are different sizes. I have never tried that before. If you turn off snap and turn on the mouse, if you do small movements does it work? Then when you move further goes strange? Or is it one eye moves faster than the other? Understanding the rigging and layers like "pupil range" layers makes a big difference.
For triggers, there is a triggers panel in rigging mode. If you look through the swap sets and triggers, are any of the bindings orange? That indicates that one is broken. You can drag the new layer over the top of the existing trigger (then delete the orange broken link). Doing in that order will allow you to keep the existing trigger bindings and swapsets (if that is what the problem is).