Longe on a deep surface, for the horse.

It's stressful enough on the horse's body (& mind) to circle much, yet it's much harder still if longeing takes place on a hard surface. The weight gets concentrated on the insides of the hooves as the horse leans to the inside, and deep footing cushions this.

However, if it's too deep it's doing more harm than good. Deep footing pulls on the leg and makes it more difficult for the ligaments/tendons. It helps the horse to go in straight lines as much as possible, and keep the circle as large as possible. Doing transitions often and varying speed will help keep their minds and bodies working better. If you understand how to take a horse around a circle properly, you do not allow them to lean on the inside shoulder in the first place and thus reduce strain.

Roro, please share how to prevent horses weighting the inside edges of their hooves on a circle.

Same thing you do under saddle to keep them off the inside shoulder, really. Don't let them turn their head/neck in and keep their hind moving, walk towards the barrel to press the shoulder out. Their neck should be relaxed and reaching and not bending either towards or away from you. Like I said, going in straight lines when possible will help them from getting 'stuck'. If they're still leaning in, chances are you need a larger circle and to put more pressure on them. Obviously it won't be exactly the same as going in a straight line, but getting the horse to move correctly will help.

The hoof will always be parallel to (flush with) the ground surface, & so the tighter the circle, the more the breakover point shifts from the center of the hoof to the inside edge of the hoof. The tendons, ligaments, & joints on the outside will be stretched to accommodate the turn, which stresses the horse's body because it naturally has very slight lateral stretching capacity. Barrel-racers have very deep footing for their tight turns, race horses have sufficient depth & spring in the tracks, yet longeing is often done on insufficient footing, & done for too long a time, as well. The increasing instances in stifle, navicular, etc. nowadays are the result of humans mishandling their horses in longeing and other ways under saddle.

Yes, I realize this. If you read my post carefully, you would have realized I was giving advice on how to lessen the pressure, not denying that going in a circle puts more pressure on the inside of the hoof. Words like 'often' and 'mishandling' are too vague for me.

All I have for all my horse work is a paddock with sparse grass, less in winter. So I guess they're going to have to deal.

I honestly believe that lunging on grass is better than lunging in too-deep sand/footing. Horses are practically made to live on that kind of ground, not 8 inch deep sand that pulls on their leg when they lift the hoof up.

Their anatomy dictates going straight, mainly; the time spent on circles is imposed by humans.

Tell me Northern, how deep of a footing are we talking here?