Don't be afraid of speed work (for healthy horses)
Slow, postural work should be the basis for a balanced horse, but faster gaits have their benefits too.
Canter or gallop can be a great way to mobilize the lumbo-sacral junction, and putting the large powerful muscles to work.
that he learns the correct technique when it comes to carrying a rider. Slow work is geared more towards educating the nervous system and creating strength in the core musculature.
And while most riders brush over this part and progress to faster work too soon (to the detriment of good posture), some riders get stuck in the slow work and seem to be afraid to work in faster gaits. This can hinder the horse's progress and can lead to problems down the line. There are many benefits fo working in the faster gaits, especially under saddle. For one, riding in walk alone, can actually have a negative impact on back muscles. More on that here https://www.thefunctionalhorse.com/blog/riding-in-walk-only-friend-or-foe. The big movement muscles, like the long back muscle, the gluteals and neck musculature need to be put to work. While small postural muscles benefit from slow, steady work, the big muscles do better wehn they are asked for more power. This will contribute to developing a nice topline.
And it's not only muscles that benefit from faster work. Trot and canter have a different effect on joint range of motion than walk. The lumbosacral joint especially moves very differently in canter than it does in slower gaits. Because the pelvis tucks under way more in canter (as both legs are being pulled under the body), it creates flexion in this joint. This can have a very beneficial effect on the musculature around the pelvis (if the horse doesn't have any dysfunction in this area). A fast canter or even a galop (when executed safely) will have a beneficial effect on muscle development. Another thing to keep in mind is that the spine is way more stable in trot than it is in walk. This is great for training the stabilizing muscles. Each gait has beneficial effects on the horse's body and it's this variety that creates a well-rouned athlete.
It can be hard to know when the horse is good enough at maintaining a healthy posture that it's safe for him to transititon into faster gaits. Ideally, the horse should be able to maintain good posture all the time, but in reality, this can be tricky. It's ok if the horse takes a few steps in a less than ideal posture – afterall, he needs to experiment and learn somehow. A general rule of thumb is that a long as he spends more time in good posture during a training session than in a bad one, we can make progress. If your horse struggles with good posture in faster gaits, work in shorter bouts and larger circles or even straight lines. This goes for canter or gallop work especially. Going really fast on a small circle is never a good idea, so a straight line is preferable. And of course, always remember that safety comes first. I'm not saying you should go all out with a horse and rider that aren't used to it. Make sure you introduce faster gaits on good footing and gradually - maintaining safety.