A way I teach my horses flying changes is I’ll put a fairly small jump in the middle of the arena and as I cross the diagonal over the jump I ask the aid and the horse automatically changes rein... I’ll start putting the jump lower till it’s a pole and by the my Horse gets the hang of it... just make sure your asking the correct aids..
Start with turning across the diagonal at a canter, bring him to a trot for a few strides and then ask for the other canter lead. Slowly work that down from 3 trot strides to 2 and then 1. Finally when going across the diagonal don't bring him back to a trot but just ask for the other lead. It may take a few tries but in my experience he should do it