I personaly use very successfully Chiro/Accu on my horses, although I use one of the US Team Vets, and will not use others. Have even asked my lameness Vet to recommend someone he has suggested that I do not change.
So, Do absolutely looked for a QUALIFIED/Accredited Vet. Even though horses are big, they adjustments shouldn't be. If they use malets or really shove the horse around it's not being performed correctly.
Now, before you try hill work and strength training a few things don't equate.
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cavalettis are difficult because he has a hard time picking up his feet and bending the hock area Trot- halt- back-trot seems to help
Your trot/back-transitions are alot harder and put much more compression on the whole hind end, especially the hock. So, if it is joint related he should object to or not perform that.
Cavaletti work, does increase the flexion of the hock but only mildly. What it puts more stress on is the Sacrum. When the hind leg draws upward you create a concave compression on the vertebrae.
I would be more inclined to look higher than the hock.
If you are dealing skeletal, no matter how much strength training exercises you do, he will not get better. And certainly may get worse, since you will be building muscles incorrectly.
He must be straight in order to perform correctly.
If, You decide to try Chiro. You do not need to have it done, you can ask for an evaluation only, and go from there.
I am so against injections, they are not the cure all they are made out to be. And you can only inject so many times. 6 is awful young to start.
Other things to look at, his feet. Trimmed or shod make sure they are not out of balance. Try wearing a sneaker and a high heel at the same time. Foot wear will tell you alot. When he is due look/ask your farrier to explain his wear pattern. His feet should be long, but the same length on the front/sides-heels grow slower. If he has say an inside wall worn down, he is not loading[weight bearing] flat to the ground. That is a starting point. Crooked in feet,fetlock,stifle, hip, etc... Why is he landing crooked.
Then, tack/rider although probably not as he does it on his own.
Do check for Nuerologic issues-Wobblers syndrome comes to mind first. But EPM, West Nile and Lymes are also possibilities.
These little problems are when it would really be nice if horses could talk.
Keep us updated and do be careful.