I think the real difference between gym hopping, even once a year, and classes changing once a year, is the long-term plan in different gyms is not the same. Different gyms are going to have different "development plans", different philosophies, different schedules for skill aquisition and the like, while all of that is pretty consistent within a school. I.E., you do English 9 this year, English 10 next year, English 11 year after, and take one of several offered courses the year after. Yes, there are level progressions in gymnastics, but each gym handles them differently.
I think this is probably the most significant difference between gym and school, considering that in a school with more than 2,000 students, like mine, it's pretty likely that you're not going to know most of your classmates one year to the next. And most of the time you're going to have completely different teachers except in a subject like music.
As for gym hoppers, I haven't known of any in the area, which is possibly because it's pretty rural out here in the Midwest so the drive would be a big issue if you wanted to change gyms. Some of our girls have switched to a newer gym in town, but I don't know if that'll last...their equipment isn't as good and they're going through a coaching change, but they have ties to a university team which might look good for parents of kids considering a scholarship. But that's only one gym switch.
This is actually kind of personal for me because one of my teammates who I've had a love/hate relationship with for several years recently switched to a gym which doesn't perform as well as our gym, especially at the optional level. This girl has been struggling for years with the basics, and was working level 6 at our gym, but switched because her mother felt that our gym was not dealing with her needs sufficiently. To be honest, I think perhaps gym was just not a good sport for her. She didn't always apply herself unless there was a level change at stake (esp. in strength)...but I don't think being in a gym that doesn't perform as well will put her "on par" with the other girls.