As a vet I find it surprising how many horses haven’t been taught how to trot-up correctly. This is a valuable skill both for competition and in case of lameness. You can’t just expect a horse to understand how to trot smoothly and straightly so put a bit of practice in. I’ve nearly failed horses at 5 stage vettings which looked unlevel just because they were being dragged along and were not trotting freely and evenly. On the flip side I had a horse of my own who was nearly spun at the first trot-up at a 3DE for the same reason. Treat it like any other aspect of your horse’s schooling. You don’t expect them to be able to do a perfect shoulder-in or counter canter in the dressage phase without having taught the movements at home.
Ideally the handler should run at the horse’s left shoulder with the horse on a loose rein. Try to coordinate your steps with those of the horse so that you both move your left and right legs at the same time. This can improve the picture and prevent you detracting from the horse’s movement. Always turn to the right (away from the handler) at the top of the strip and make sure that the horse travels at an even pace in a straight line. Practice makes perfect and you’ll be grateful that you put the effort in when you come to sell the horse, arrive at your first 3-day or, heaven forbid, need to have a lameness work-up done.