In Stevens On Stowage, there are Army Regulations from 1857 on how horses should be shipped.

Basically step 1: Tire them out.

Step 2: horses should be kept from food and water in the hours prior to boarding.

Step 3: Take off shoes on the rear feet

Step 4: Use a belly sling and breast sling to raise the horse from land into the stable area of a ship already padded with lots of fresh straw

Step 5: As soon as the horse is solidly on its feet, immediately take off the sling.

Step 6: Feed only oats and grains sparingly the first few days on board. Return to regular feeding amounts but maintain a daily oat/branmash supplement.

While on board, horses are to be secured only with a canvas collar with two ropes, one per side. Horses should not be put into slings less than a week after setting sail. When placing a sling around a horse, it should allow the animal to relax into the sling, not have it actively lift the horse.

Four extra stalls should be left empty to rotate horses for daily stall cleaning. During rotation, a horse's feet should be thoroughly cleaned. Horses should also have their legs rubbed daily. Fresh air is a must to prevent respiratory disease, so a sail was pointed into the stalls for constant fresh air.

During calm weather horses should be slung. Horses will allow the sling to take their full weight while they sleep during calm weather. In rough seas, horses should be allowed to stand unslung. Horses naturally resisted rolling ships and shifted their weight accordingly. If the weather is exceptionally bad, grooms should stand at the head of their horses to help calm them.

I would guess two of the major hazards would be broken legs from kicking or losing footing in storms and colic from lack of water intake.

"Fresh" food isn't really necessary for horses. Lots of folks who own horses feed bailed hay in the winter when grazing isn't an option. I would think it would be the same standard.

Fresh water was probably the biggest challenge. Horses need anywhere from a half to a whole gallon of water per 100 pounds just to keep their bodies functioning properly. So a 1,200 pound animal, you're looking at 12 gallons of water per day just to keep the animal in minimal condition. Not enough water means food cannot pass through the intestines correctly, leading to major problems.

The 1979 film production of The Black Stallion depicts transport of the titular horse across the sea in a soon-to-be shipwrecked ship. The sling apparatus is clearly shown. Air transport is much the same.

The animals' feet are allowed to touch the ground, but the torso is partially suspended to relieve pressure and minimize the effects of standing in place for days on end in a rolling ship. Large animal veterinaries use the same set up for surgeries and recovery.