I would assess the quality of your canter. If you horse is not uphill, rhythmic, straight, and balanced, he will have a lot more trouble doing flying changes. Do some flat exercises, poles, counter canter exercises, etc. to make sure you have the canter you need. If you have a horse that runs through their lead changes, it usually means they aren't strong enough to do them without speeding up.
If you have a good trainer at your disposal, this is a great thing to work through with eyes on the ground.
It could have a lot to do with balance and strength. My OTTB transitions beautifully tracking left, but not to the right. We are working on balancing him with a lot of work to the right, stretching, and massage. Good luck!
If your horse is a bit clumsy it might be because he's lacking the balance and coordination perform a lead change. I don't ride english but I think that working over ground poles at the walk trot and canter would be a great way to get your horse more prepared for a lead chnage. You can do some pretty cool things with ground poles that have awesome benefits for your horse. Just Google 'Tristian the Wonder Horse' and watch the video s.
What the aid is may be told to you, but every rider will practice this aid with a unique variation and a horse must learn to respond to their rider's command. The most important thing in preparing for the changes is getting the canter/walk walk/canter transitions perfect! You need to feel when to ask.
Maybe your action with your right leg isn’t enough important. You need to close your right leg on the girth to ask your horse to go forward during the change.
I hope this helps.