This is very simplified/abbreviated but might help.
How a horse moves is a sum of its parts. Look for disharmony. Look at their body as a whole, not just one limb.
When I watch a horse the first thing I look at is their general shape and back. Is their back tight or hollow? How is the neck and head in relation, does the neck look pulled back and is the head inverted? The neck acts as a rudder to the rest of the body, its relationship with the body as the horse moves is important.
Is there overtrack behind? Overtrack is when the hind hoof lands just in front of the front hoof print. Not all breeds over track so you need some familiarity - and some breeds naturally overtrack by a lot (gaited breeds in particular) so it is not always a good indication they are sound. But look for differences between stride length.
Watch their hind end as they trot away - does one pelvis tilt, or is one hip higher or lower? How about when standing still? If you visually put a plank on their croup would it fall to one side? Most of the time hind end lameness is more obvious when watching them trot away.
Study the natural range of motion (ROM) of front and hind limbs. Because your eye depends on spotting deviations. Is the flight path of the limb straight? Or does it behave abnormally during abduction/aduction phases? Does it slam down abruptly and offload? Does the hock twist or wring out?
How do they react when asked to move off? Do they jump into a trot? Is it fluid? Or short/choppy? Does the head go up or down? Does the stifle or hind end slip in downward transitions?
Remember “down on sound” for front limbs (AKA the horse’s head will jerk UP when landing on the sore limb), but the opposite is usually true for hind limb (although hind limb lameness can be all over the board).
Traveling forelimb lamenesses are usually related to neck or hind end problems.
Watch the hoof fall too. How a horse moves over their feet can tell you where the lameness is. A horse that flicks his hind feet but won’t push off? Probably heel or foot pain in his hind end. A horse that lands very flat and you don’t see much ROM at all up front (aka in economy mode) may have bilateral footsoreness. Tight shoulder and not much neck movement might mean foot sore up front but it could also be them stabilizing a sore back.
Watch how the tail behaves too. Stiff tails held away is a sign of a sore horse. Could be SI, could be back or suspensory. Abnormal tail carriage to one side is usually SI or hock. A tail that is clamped down, with the loose tail hair being jostled between the hocks as they trot, is usually hock or hind foot pain.
In your own horses, observe how they behave in the pasture. Do they run often or do they tend to not move at all? Do they walk uphill willingly? When cantering do you see fluidity in the stride or do they bring their hind end down fast and quick each stride? Can they stop well or is it a mess? Do they voluntarily avoid gravel or insist on walking on grass leading to/from paddocks?
Rather than zeroing on one leg it’s all a picture; the entire body is connected, horses cannot separate their sore limb from the rest of their body and that compensation is sometimes more obvious than the limb itself.
