Your question covers several thousand years, several score cultures, and a lot of acreage. So I'll trim it back to what I know from a ms I have near completion.

Throughout the history of the world, the most common transport animal is the human. They are excellent for long distance and fast travel, with better endurance than most animals, and wherever you go, you can find some, starting with yourself.

Have you seen the marathon performances of East Africans? What do you need that's faster or more enduring? The Andes kingdoms relied on fleet-footed messengers as did the ancient Greeks. Check where the idea of the marathon comes from - messenger who ran from Marathon to Athens with news of the battle just fought. Even the horse-oriented Mongols used runners. Yes, they had their "arrow riders" galloping from one remount station to another, eventually handing off their bag of messages to a fresh rider (an idea from the Persian Empire Herodotus wrote about). But in areas that were in some way bad for fast horses, rather than pick a way for hooves, they gave it to an arrow runner. Both runners and riders wore bells, so the sound would let stage stations know they were coming in.

The Horse in West African history covers just that, but only from medieval sources forward. So we know they were used in Sub-Saharan Africa, both for transportation and cavalry. But that ends at the tsetse line. Horses never adapted to sleeping sickness like humans did. There's also the Dongola horse that came down the Nile. In times we would call ancient, the Garamanteans of the Sahara oases used an original form of horse traction for chariots. Spruytte covers this.

Camels came, too, in the trans-Saharan salt caravans. There's a good NatGeo article on the modern salt caravans, still on camel. But we find them embedded in life all across the Sahel, the dryish lands south of the Sahara.

In the tropics and below, porters were much used for moving goods when rivers were not available. The regulations for their use in Africa in the 1930's, from Osa Johnson, match the weight limits for Burmese porters in the 1890's (Holt), making me think this is a pretty universal limit for professional porters: 60 lb/27.2 kg apiece.

In the late 1800's there was an attempt to domesticate zebras in South Africa, but the rise of the automobile shelved that. They could be taken as foals and tamed, and went well in harness. Riding could be another matter, as they have short stiff necks, which make them hard to rein. The first doctor in Nairobi, though, rode one on his rounds. Hybrid zorses and zonkeys worked out pretty well, too.

But, no, no ancient African people show any sign of having domesticated the zebra. Such a missed opportunity.

Also, in the Cold War, the Soviets tried domesticating elands for their African allies, but that didn't work out so well.

In South America (and for that matter, Central America), porters were the baggagers, usually using a tumpline over the brow and back to the load. They were also the swift messengers. In A Book about Travelling, I got an engraving of porters carrying a European couple, about 1840s by her dress & coal-scuttle bonnet. They are seated on legless wooden chair strapped on the porters' backs.

Of course, before Europeans, the royal and noble might be carried on palanquins.

But in the Andes, the camelloid guanaco was domesticated as a llama, and some will tell you they're the best beast of burden. Toward the end of the 20th C they were imported into the mountain West of North America for "llama packing." Hikers could carry more food or comforts with them while leading a pack llama. They are generally a placid and cooperative animal that grazes along the trail and requires little upkeep. They don't need grooming. Their thick fleece acts as the pack saddle: throw a balanced load on and strap it down. "Camel of the Clouds" from NatGeo is one of my primary sources, besides llama packers. There are horse-packing pros that had used mules for baggagers that have gone to llamas just to make life easier on trips into the Cascades.

Llamas do have the peculiarity that if they feel overloaded, they just kneel down where they are and won't move until you lighten the load.

SOME SOURCES:

Dell, Anthony; Llama Land East and West of the Andes in Peru; 1927; George H. Doran Company; http://www.archive.org/details/llamalandeastand013611mbp

Carles, William Richard; Life in Corea; 1888; London; New York; Macmillan and Co.; http://www.archive.org/details/cu31924023275641

Hodge, W H; "Camels of the Clouds"; National Geographic, May 1946

Johnson, Osa; I Married Adventure: The Lives and Adventures of Martin and Osa Johnson; 1940, 1942; Halcyon House, Garden City, NY;

Kitchiner, William, MD; Traveller's Oracle; Or, Maxims for Locomotion: Containing Precepts for Promoting the Pleasures, and Hints for Preserving the Health of Travellers. Part I. Comprising the. The Whole Revised by William Kitchiner, M.D., &c. London: Henry Colburn, New Burling­ton Street. 1827; http://books.google.com/

Knight, William Henry; Diary of a Pedestrian in Cashmere and Thibet; PG; Himalayas, Tibet, Cashmere; 1860; http://www.archive.org/details/

Law, Robin; The Horse in West African History: The Role of the Horse in the Societies of Pre-Colonial West Africa; Oxford, New York; Oxford University Press, for the International African Institute, 1980

Llama Products; March 28, 2011; https://www.llamaproducts.com/index.php?page=articles.tips

Spruytte, Jean, translated by Mary Littauer; Early Harness Systems: Experimental Studies; A Contribution to the History of the Horse; 1977, 1983; J. A. Allen, London