My stepmom has leash trained her kitties...but it is only the extent of her front yard so kitty can have a jaunt in the great outdoors.
Are you wanting to leash him for the car ride? (that is how I'm reading your post).
If so--he is much safer wailing in the cage. In the car, your cat is going to seek cover and hide--and though you may be holding the leash, or have it attached to something--it creates an unsafe environment for all in the car especially the cat.
The vet we go to, we have to keep the cat on a leash.
With the dogs that visit the vet, I'm not sure that your cat would be any happier on a leash.
Good luck.