I just got curious, what animals were used by Asians (peasants and rich people) for riding in the Middle Ages? In the west region of the Earth people rode horses, but I guess horses were not common in Asia, or am I wrong? Thailand/Vietnam/Laos/Cambodia... this region interests me most.
Given that the stirrup was invented in Asia, horses were not a rarity in Asia. Look at Asian art work, and you'll see that horses were not rare at all, but common.
What animals were used in Thailand, Vietnam, Cambodia and Laos? Horses were common riding animals. So were donkeys and mules. The most common draught animal was probably the water buffalo. Even today a lot of Thai farmers use water buffaloes in their fields. More so in lesser developed Laos and Cambodia.
I'm not too sure about elephants. They certainly weren't common riding animals. Catching, training and keeping elephants is difficult and expensive. In all southeast Asian countries you're interested in royalty was associated with elephants. Elephants were used to haul heavy items, but definitely weren't used as common riding animals. Only royals and those approved by royals did that.
In Thailand, which is relatively flat (at least in the plains) with lots of rivers, riverine transports were very important. Thailand still has many canals. They were used for transporting large and heavy goods, and for transporting troops. Many canals were not dug for civilian purposes, but because of military needs.
I don't know if this also applies to the medieval period, which roughly translates to the periods of Sukhothai and Ayutthaya. After the fall of Ayutthaya, when king Taksin regained the kingdom, many canals were dug on his command to make military transports possible. I assume that this was also the case before the fall of Ayutthaya, perhaps to a smaller extend.
In any case, riverine transport was the mode of transport for anything. Most cities didn't have roads, they had canals. Every form of long distance transport was done over water. Only when this wasn't possible, animals were used.
Jeremias van Vliet was the local (Dutch) VOC director in Ayutthaya from 1629 to 1634. It took Jeremias 30 days to travel from Bangkok, which was a small port at the time, to the court in Ayutthaya. He used various modes of transport, from walking via oxcarts to elephants, but mainly he was transferred by boats and barges. Using the same route, it took us 4 leisurely hours in an air-conditioned coach.
Horses were omnipresent in Asia at the same time periods that we call the Middle Ages in European history.
In the 13th century, the Mongols were riding roughshod over all of Asia, and given that they were probably the best riders in history (certainly of the time), any area they conquered (all of China, much of the Middle East and India) can be assumed to have the horse as a major component of its society.
In regards to Southeast Asia, wiki states:
The Yuan Chronicle reports that only 700 men defeated a Burmese army of 40,000 to 50,000 with 10,000 horses and 800 elephants. It also reports only one Mongol was killed, in trying to catch an elephant. According to Marco Polo, the Mongol army consisted of 12,000 mounted archers, and the Burmese army numbered 60,000 men with 2000 elephants,
So we can see that the Burmese at the time used both horses and elephants.
Another major country the Mongols was unable to conquer was Japan. According to wiki again the horse arrived in Japan around the 4th century AD. It was certainly a major component of Japanese society in the middle ages.
Of course, jungle terrain is far less conducive to the use of horses than the open plains of Mongolia, so you likely saw much less usage in places like Laos, Cambodia and Vietnam, but the animals were certainly known throughout the region, given that the horse is one of the animals in the Chinese Zodiac and that the entire region of Southeast Asia looked to China for much of their scientific and philosophical knowledge.