Don't ask him to speed up, you'll teach him to rush his changes.
Don't practice them in a slow canter, however... if you ride with more impulsion, it will be easier for him to change.
If it's not a problem of pain then it's one of two things--doesn't have the strength, or he's not supple enough.
To make him more supple, ride him at a counter canter (let's say, right lead going around the arena to the left), and practice keeping his head straight ('neutral'), and then bending his head INTO the arena (into the left). If he has a terrible time of doing this, you'll know he was stiff. Once you can, it should help change his back end.