As stated, donkeys are rather small animals, and usually when used with a cart are used singularly. A pair is occasionally used, but not as commonly. The carts are rather small, and the maximum weight is not high. The driver walks beside, as the driver adds weight that reduces the payload ability. I'm not aware of donkeys being used for long, multi-day hauls.
Mules can be much larger than donkeys and would be the preferred choice for larger loads and longer distances. Mules work better in teams than donkeys. The largest mules can weigh in excess of 1200 pounds and rival draft horses in pulling power. Mules used to be bred in a wide variety of sizes to meet the market needs. Mules tend to walk faster than donkeys.
Horses are larger than mules, and ultimately, have more power. However, mules tend to have more endurance than horses and can go on longer. Mules also need less food than horses. Mules at high levels of use needed a bit of grain, with horses needing a fair amount of grain.
Oxen are comparable in pulling power to mules and horses, but are slower. They tend to be single gaited, and you pretty much go whatever speed the oxen want to travel. However, they keep on going. Their hooves also tend to grip better than horses. Oxen can survive on graze, spending the evening grazing and the daytime chewing their cud.
Oxen are more rugged than mules, which are more rugged than horses. Oxen are also easier to maintain. Oxen were the preferred choice in the American West for transportation, but I don't know the numbers. Mules were preferred over horses for pulling, but frowned upon for riding.
Cross-country travel was mostly done with the driver walking. A person weighing 150 pounds isn't an insignificant weight. The picture of a team of horses pulling a family across the prairie is a joke. The family walked most of the time and horses weren't used.
Oxen and mules in the same wagon train don't tend to work well. The mules walk a lot faster.
Daily care would be brushing and feeding. Horses and mules would need their shoes maintained, although mules especially, depending upon the terrain, might not need shoes. Water is a major requirement in some areas. When deserts are crossed, frequently the last few miles were done at a gallop, as the water-starved animals raced to the water source.
For rugged roads, I'd probable go with solid-wheeled carts, as they suffer less breakage then spokes. The wheels, however, do add weight to the detriment of payload. Extra wheels are a requirement, and although an axle can be easily made from a tree, I'd probably carry a spare. Jacking a wagon can be done by inserting a pole under the wagon and everybody lifts. Then rocks are inserted to hold the wagon in place.
The choice between mule or oxen is less clear-cut, but I'd think I'd prefer go with oxen. Much more reliable over the long haul.
Best of luck,
Jim Clark-Dawe