Gracefulness on the range isn't a function of one's natural ability to move beautifully. Which is fortunate since I'm a woman who has been known to trip over perfectly flat floors and walk into large, unmovable objects. However, over time, I've been able to integrate movement with my shooting and look pretty doing it because it's a skill that can be learned.
When you shoot an action sport, you will often either have to shoot while you are moving, or move between shooting positions. Shooting on the move requires two things: solid fundamentals and practice. If you cannot consistently make your hits when standing still, it's unlikely that you will be able to do so while your feet are going. Work on that first, then start adding movement. There are a lot of techniques to improve accuracy while you aren't on a stable platform. I encourage you to try each one you are exposed to, then try them again. What works for you with pistol might not work when you're shooting a rifle, and what works for you today might not be the best option for you in six months. Once you've found something that seems to get decent results, work at it because shooting on the move can only be improved by shooting on the move. Most shooting on the move "tricks" are intended to make movement as smooth as possible, which is why the particular one you choose doesn't matter so much as nailing down the technique that makes you move most smoothly. Jerkiness at the feet translates almost directly into jerkiness at the sights and inconsistent groups.
Moving between shooting positions does rely quite a bit on your physical abilities but no matter what your physical limitations, you can still benefit from working on moving as efficiently as possible. And don't forget the finer movements inherent in shooting like drawing a pistol, racking slides or bolts, or loading ammunition into a firearm. Efficiency matters with that type of movement too, and the key there is to find a consistent and repeatable way to perform every action. While a specific way of locking back a slide/bolt may feel awkward at first, with practice you will find the specific movements that work for you every time and they will feel more and more normal. Much of the grace you see in top shooters comes from the fact that they don't waste time and energy with extraneous movement. Instead, they pick the shortest line between two points and the most natural movements to reach for and manipulate their gear. Focus on finding those efficiencies and your shooting will also become more graceful.
Don't believe me? Use one of the favorite tools of the moment and have someone take video of you shooting a few stages, then do it again in a few months or next year after you've worked on making your movements smoother and more efficient. Watching yourself from a third person point of view will help you evaluate yourself more neutrally, instead of relying on your memories of every little bobble and stumble. And as you shoot prettier, you'll find as I did that learning to move gracefully on the range is also learning to shave wasted time off your scores.