I attack the gymnastic movments in several ways.
One or two days a week, I do mostly gymnastic drills. Also, I nearly always have one element that I will practice multiple times daily in grease the groove fashion. (people at my work are finally getting used to me doing a quick press or planche here and there)
I tend to do the tumbling more on off days as active recovery, but this is an area I am plannning to focus on more - it needs work!
During the training sessions, I work thru progressions on whatever skills that I am presently training, but I also concoct mini WODs with some of these elements. I get a little met-con this way, and build some strength endurance. However, I don't limit myself to only the gymnastic elements here. I really like doing o-lift work or squat variations mixed into these circuts. Movement and mobility sequences are another great option.
Here is an example. If you have several skills that you can string together on the rings, design yourself a little routine. It might be something like Muscle-up, L sit 3" hold, press shoulderstand 3", fwd roll to hang, to back lever 3", straddle front lever 3", 3 L pullups. Then, select a lift for the day. Here, then, is an example WOD:
Squat clean X 3
High box jumps 36" X 5
Ring routine X1
The routine provides a potent stimulus that delivers a training response greater than the sum of the individual parts. I strongly suspect that adding the non-gymnastic elements amplifies this effect even more.