Lunging a horse is exactly the same body language and cues that you would give if you were doing join-up with a horse. You are driving the horse in a circle and keeping them going when they slow down. If you have a respectful horse and you lunge them for long periods of time, you will notice them lowering their head and licking and chewing, exactly the same as if you were doing join-up. So if you completely ignore this and keep them going, eventually they will stop doing it. It is something to keep in mind when you lunge, it is NOT the same as riding because you are guiding the horse in different directions and the horse's attention is in front of him, not beside him while he runs in a mindless circle for ten to twenty minutes (unless you just rode your horse the same direction in a round pen). It is about as mindless as running on a treadmill for an hour without knowing why you're doing it. Lunging has not been around for very long either, if people wanted to exercise their horse, they would ride them or take them on walks. Lunging is a good way to help a horse release built-up energy or to gain respect on the ground. Yet recently, people have also discovered how convenient it is that you can stand in one spot and have a horse run in a circle around you at any gait, so why not use it for exercise?
Lunging a horse has been around for many many years. It is not a recent invention. I have no idea where you come up with the things you do. And where did you get idea that people took them on walks to exercise them in olden days?
And it is NOT the same as "join up". It is done for suppling, energy release, getting kinks out, to get horse in the beginning stages of being trained, prior to training to cart, or to saddle.
And not done for long periods of time either. Done long enough to get done what is needed. No need to overdo this.
As for OP.
It could be that horse is tight on right side when going to left, chiro or massage would help possibly with that.
If you run your hands over his body, very lightly, you may be able to "feel" an area where body is tighter, which could be area causing trouble. Think how your shoulder muscles feel when you sleep wrong, that is what you will be feeling for.
Could also be, if this horse is being ridden, that the saddle is leaving sore spots too, due to issues with saddle? Can't remember if under saddle yet, will go see, but something to consider.
Is he doing this at all gaits, or just one?