I have a super thoroughbred in training now that recently came off the track. As you can imagine with any thoroughbred that has raced, it takes time to teach them to strike off into the canter off the hind end instead of falling into or racing into the canter off the forehand. It also takes time to teach a horse to immediately lift off into the canter instead of "chasing" the canter depart. Your canter transition is really important to do correctly because there is a specific score for the quality of the canter depart in many dressage tests.
So, what does a good canter depart look like? Regardless if it's from the walk or the trot, it needs to be round, SOFT, and submissive. The horse's head should stay down and round (but not forced) and there should be total relaxation and a smoothness. To achieve this, your horse is going to need to step off with their hind leg into that depart.
What does a BAD canter depart look like? It looks forceful and aggressive - and chasing. If you are chasing your horse to canter for more than one stride, then just go back and reset and try again, because that is not a behavior you want to reward. The difference between a good and bad canter depart is like ballet vs. kickboxing. If it's a bad canter depart, the horse will throw their head up and invert, lose their balance, and probably race on their forehand a few strides before they "fall" into the canter through a loss of balance.
But remember this is dressage, so we need to make it look soft and graceful.
So how do we "fix" a thoroughbred off the track so they have a neat and tidy canter depart?
Here's what you don't do. First, you don't fix your hands and make them round from the front end to force them into it. I mean, if you're a skilled rider you can get away with that and probably fake it in a sales video and at training level, but it's not going to TRULY help you or the horse advance up the levels and do correct dressage training. You're gonna be screwed by the time you get to second if you keep trying that!
Now, you need to isolate and strengthen the hind leg. This is assuming you can bend your horse laterally side to side without their shoulders moving, also assuming that you can move the shoulders with or without changing the bend, and that you can also isolate the horse's hind legs. That's how you make a straight horse right? And when you have a straight horse you have a round horse!
Assuming we have some of those flexibilities and we don't have to muscle the horse into it, I like to set up young horses to do canter departs on twenty meter circle with two eight meter voltes in each side. I use these voltes to help set up the bend and as I ride the volte I ASK for the depart as I am coming out of the turn and in towards the arena fence. Then if the horse picks up the correct lead I canter out large on the twenty meter.
Here's the key point to make a delicious canter depart on a young horse....it's easier at first if you do it out of a shoulder turn or counter bending turn. By using your reins and leg to move the shoulders around the circle, it separates them from the hind legs, and then when you apply the aids for the canter depart, the horse is in the perfect position to strike off. The shoulders are lifted from the shoulder turn so the horse has to step with the hind leg, and there you go...a perfect canter depart from the hind leg. It might take a horse a few tries to figure it out. Just reset on your pattern each time and ask again until your horse figures it out. In a horse with a weak hind leg, no amount of forcing the front end is going to fix that strike off, but lateral work while controlling the shoulders will help strengthen the hind leg and create a beautiful canter depart.
Other tips if your horse isn't getting it:
Stop throwing away or giving too much with the reins (but don't hold too much either - the horse should have it's head and neck in this exercise so they can figure it out)
Stop leaning forward
Stop holding too much
Make sure your legs are in the correct position
Make sure you have a good shoulder turn with GOOD mobilization of the shoulders first
Stop going off course and the structure of the exercise
If your horse really doesn't get it or you are constantly getting the wrong lead go back and work on the hind leg.
If you need more help you can always set up a virtual lesson.