I transported my own horse for the first time when I bought her, I normally pay others to do it but she was too far out of the way.
There was a few things that I was told. Try not to stop - like if you're heading into a roundabout slow right down and try not to have to stop and start all the time. And apparently the thing that really unbalances the horse is like the slope of the rode (how it slopes of to the sides) so, if you can, avoid it, like keeping near the middle of the road etc. Also don't go too fast, take corners slowly. A bad driver can make horses really bad to travel.
I'd probably check on them at least once. I had a horse that used to always fall down in the float
So I get a little nervous.