I'm still around! I can't keep up with the daily stuff with a new baby, a toddler, working full time, and all my critters but I do check in from time to time.
Great question! While no one has studied the cleaning of the eye in real time with something like an MRI to see exactly what's going on in there I think I can shed some light based on eye surgeries and necropsies I've performed. The eye ball itself is round for the most part but the iris and part we can see is more conical compared to the rest. There is also a membrane called the third eyelid that humans lack but most other animals tend to have. You've probably seen it on your dog or cat. Interestingly I can't say that I've ever seen it on a necropsy...will have to look for that next time. But I've seen it come up in chameleons enough to believe it's there.
So when they need to clean their eye in their weird way they can basically suck the eyeball deep into the turret and then the third eyelid can elevate to push out debris and cover the opening. Part of the weird movement you see is the eyeball itself rolling around in there too. It's no surprise that a creature with that much voluntary eye movement would be able to do some under eyelid gymnastics as well. And since the little part of the eyeball we can see is more conical it is more obvious when it's moving around and pushing against the skin of the turret from underneath. Rolling around the eyeball against the inside of the turret I imagine it would be to clean off any debris on the cornea and try to dislodge any debris/mucus/etc. Then when they're done all the structures return to their normal place. I might have some gross necropsy pictures of the eyeball if you're interested...