There are many different variables at play here, and it's not just restricted to exercise routines...
Weight: It takes less effort to push a 110 lb body off the mat than it does a 200 lb body, so by having a leaner body they are more efficient with their muscles.
Practice: It takes practice to do complex moves, and flips are certainly no exception. Simply having an identical body type to gymnast does not mean you can backflip. More than anything you need to practice.
Muscle Density: I like to compare bodybuilder muscles to that of a peacock... large, beautiful, intimidating, but really just a facade. Contrast this with a strength trainer's muscles which are more akin to a leopard... lean, not very large, but powerful and deadly. Gymnasts fall into the latter category of having muscle that is small, but incredibly dense and powerful.
As for actual exercise routines? That I cannot attest to, having never been professionally trained as a gymnast. That said, it should adhere to a few basic principles:
- Functional strength. You want to build a base amount of strength that is functional. For example, low rep high intensity squats/deadlifts.
- Power. Strength is not enough to do gymnastic moves, you also need explosive power. Examples of good exercises to improve your power are push press and clean and jerks.
- Practice. You need to practice the actual movements, the flips and the tricks.
I would place practice as the most important part of the regimen and certainly requiring the most time and effort. That said, strength and power are very important as they create the foundation upon which your moves rely upon.