Donkeys were beasts of burden.
According to the Legend of Sutekh, they were among the animals associated with the titular Osiran god of evil and violence, Sutekh. Other animals associated by the Egyptian tradition with the cults of Sutekh included crocodiles, hippopotamuses and pigs. (PROSE: Background)
In Earth's history, they were often used to pull carts. Rose Tyler rode in a donkey-drawn cart when visiting the Roman Empire in about 120 AD. The Tenth Doctor, then separated from Rose, also made use of donkeys to travel.
Like horses, donkeys were often kept in stables; Romans needed to keep a vast, informal network of stables along well-travelled routes. This allowed passengers to rest their donkeys at regular intervals. It also gave travellers specific points at which they could negotiate for the sale or hire of donkeys.
At one of these stables, the Doctor said that horses were "better" than donkeys, intimating that donkeys were relatively slower. Donkeys were also distinct from mules, but neither the Doctor nor any of his friends and acquaintances gave any particulars. (PROSE: The Stone Rose)
The Miller kept a donkey in his barn in 1666. When Richard Mace and the Fifth Doctor visited the barn, an exhausted Mace asked the donkey if it was capable of carrying a tired thespian. (TV: The Visitation)
The Twelfth Doctor befriended a donkey in Blackpool, 2089. He named her Meghan. (PROSE: All the Empty Towers)
During Christmas, donkeys would be present on Blackpool's Golden Mile. (AUDIO: Death in Blackpool)
References
The First Doctor once proclaimed that "men breed men and donkeys breed donkeys". (PROSE: The Sons of the Crab)
The Mona Lisa claimed that her model "farted like a donkey". (TV: Mona Lisa's Revenge)