I am right handed and left eye dominant...
I beat my left eye into submission and forced it to act correctly. It can be trained to work, now I only close it if I am tired or not paying attention.
My right eye has an astigmatism on top of being weaker than my left. Glasses helped me a lot, I had ones made specifically for shooting and that balancing of my eyesight helps a ton. I allow the right eye to get comfortable being in charge for a while. As I begin to tire the worst that happens is I have to help the right side a bit by closing the left just long enough for the right to get the hint and then I can open the left again. I found it to a be a training issue, but an admittedly a long difficult road.
Guys who switch hands can often be successful, for established shooters, it's because that side usually has no bad habits. We always see people do better when they switch hands under proper instruction. But it's expensive to switch everything over to the opposite side.
For a new or junior shooter, you can switch them to the dominant side early, and skip the retraining and cost of switching. It's often the easier route to take. It's an effort to train the left side to work with the right.
Look into glasses, see if it balances out their vision, to me that helped keep me on the right side.