I'll try and explain the flying lead changes for you. They are an advanced maneuver that requires a balanced horse, but are invaluable in the show pen.
First, to ask for a canter in the left lead, keep your horse going straight and use your right leg on the barrel (just behind the girth - I really don't move mine back any) to push your horse's entire body slightly to the left. They should stay straight, though. It's the opposite for the right lead.
To me, there are 3 main things that you have to have to get a good lead change. The first is staightness, from the horses nose, neck, through the shoulder, barrel, and hind end. You can't have your horse leading with one body part, sticking out a shoulder, leaning their barrel on your leg, or leaving their hind end behind. All body parts have to be centered between your leg and hands.
The second thing is forward impulsion. This doesn't mean just going fast. It means the horse driving from his hind end, really reaching underneath himself with his hind legs and going up into your hands. You should be able to feel his hind end driving up and trying to push you up and out of the saddle (but keep your behind in it!).
The third is they have to get off your leg. If you have good lateral movement, like a 1/2 pass or two-track at the walk, trot, and canter, you are way ahead of the game. You don't have to be able to move sideways to that extent, but your horse has to respect your leg. Again, not lean up against it, move forward off of it.
When I'm schooling a horse on lead changes, I always use a large area with good footing. I'll give the sequence for a right to left lead change. I'll canter my horse on the long side of the arena in the right lead, really making sure my horse is straight and moving off his hind end into my hands. I'll make the first corner onto the short end of the arena, again keeping him properly balanced (they should gently bend going around the corners), and keeping that good strong stride rhythm. Going into the second corner, I'll cut across the arena in a diagnol, trying to get the longest possible straight line I can in the entire arena. At first, just practice getting your horse completely straight down that entire line. You can either counter-canter out of it, or stop him when you reach the other corner and you have to turn left.
Once he'll stay staight down the line, I'll use my left leg and move his ENTIRE body over off my left leg and to the right. It's not a big move, more of a shifting of weight, going over 1/2 a step, and just generally getting OFF of my left leg. I'll practice that a few times, making sure he'll move off of it easily.
Once he'll stay straight AND move off of my leg (practice the first two in both directions first so you know he'll get off of both of your legs), I'll make that turn onto the diagnol, get him straight and moving into my hands, use my left leg to move him to the right, and as soon as I feel that shift in body weight I'll completely take my left leg off and drive with my right leg to move his ENTIRE body over to the left. Theoretically, they should change leads if they actually move their body over. It's hard for the horse the first few times, but keep driving with that leg and don't give up. If they start leaning on my right leg, I will take that leg and give them a good thump on the barrel. One warning, though, a lot of horses will kick out or buck when learning a lead change. I personally don't get onto them for it if they actually change leads. Heck, they moved over and did it. However, if they do it at any time other than an actual change, I get onto them for it. After they learn to move over and change, then I start getting after them if they kick out or buck during the change.
The most common problem people have is keeping the horse straight. The shoulder often tries to pop out (if your'e changing from right to left, the shoulder likes to go left). If it does, then they will only change the front lead. If that happens, block with that left rein and really drive them from behind. They may change in back first - if so, just make sure that you aren't driving so much that you're swinging the hind end across and leaving the shoulder behind. If the lead change is real rough and they seem sto stutter during it, they need to be going forward more.
I hope this helps and makes sense. If it's an option, I would have a trainer help you with this, but either way good luck with it!