Heres the scenario - You spend a lot of time making the bird good as a game hawk. Off you trot onto a grouse moor, work hard to get your dog on point, cast the falcon and off he goes. Once he gains pitch, you call him over and the second you flush......a crow decides to come over for a nosey and the falcon checks on the crow. Grouse buggers off, happy as larry, dogs pissed off, your raging, falcon catches corvid a mile away, so off you trot across the heather, day wasted.
Its easier to turn a game hawk into a pursuit bird, than the other way round. I wouldnt even attempt it.