I'm talking specifically about horses here. The modern horse evolved on the American Great Plains, then crossed the land bridge into Asia.
It then became extinct in the New World (possibly as a result of human hunting). The horses in Siberia and Mongolia fared better and equines (both wild and domesticated) spread outwards from there.
The donkey, on the other hand, originated in north Africa and spread into the Middle East.
It's worth remembering that horses are cold climate animals whilst donkeys and zebras are hot climate ones. Donkeys are a desert animal - hence the ears!