Some horses align themselves behind the human as a default way of walking together. However, for some exercises like shoulder-in, it can be useful to walk at the horse's shoulder. In this post I will describe some methods that have worked for me in asking my horses to walk next to me. If you do it in other ways, please share them with us!
The fastest and easiest way for us is to simply look to the area in front of my horse and point there with my outside (horse-side) arm. This might work even better if the horse has learned to target the human's hand but also works for my horses who have had no specific target training. If you lean forwards and point further down to the ground, you can even use this to get a long and low walk with the horse's head close to the ground, which is a way of walking that automatically makes most horses move more forwards in relation to their human.
Another way is to start walking from besides the horse. However, most horses will simply wait for you to walk a few steps and only then follow you, aligning themselves behind you. Therefore, it can help to get the shoulder moving right from the start and then gradually change that movement into a forwards movement. To do this, you can stand next to the horse's shoulder and ask that shoulder to move away from you by moving your outside hip towards it - and of course reward even the slightest weight shift in the beginning. Once the horse's shoulder yields away from your hip, you can simply add a little forwards step at the very moment when he is initiating the movement, and then gradually move less sideways and more forwards. This will result in a horse who is moving forwards sideways with you, so you don't only have the position but get the bend as an extra benefit.
Many horses automatically choose to walk next to the human when they are more playful and energetic, and also when they are more in sync with the human's moves. Perhaps that is because it is just easier to perform a coordinated joint movement when being next to each other. Therefore, the third way is based on giving the horse a reason to move in that way. I do this by frequently changing the speed and direction of my movements. For example, I walk a few steps, then stop, turn to one side and walk into the new direction for a few steps, start trotting, switch from trot to standstill, and so on. However, I make sure that although these changes can be quite abrupt and unpredictable, I still initiate them slowly enough to be able to see whether the horse will follow. If he does not, I stop and return to the way we were walking before, because a too abrupt change can result in the horse just staying where he is while I am moving away, so the connection is broken. I imagine it as a rubber band between me and the horse that can be stretched a little but that pulls me back to the horse if it is stretched too much.
Very soon the horse will realize that it is easier for him to do these changes with me (and get rewarded for them a lot) when he is next to me. Once he has learned that there can be frequent changes in our joint movement and is anticipating them, I reduce my cues for the sudden forwards moves. That is, I don't actually do them but just pretend to, which gets the horse to do a small version of them but not really start running either. Instead, he is just moving a tad more to the front, which I reward immediately. It's a bit like flipping the horse forwards by swinging your hips forwards in walk. More energetic horses will do a real jump, whereas slower horses will just increase their walk speed a bit. Thus, I adjust the energy of my own movements accordingly, so that I get exactly the response that I want.
Looking forward to reading about your experiences with asking your horses to walk next to you or even in front of you.