Did you know that almost all of the mammals that we think of as being native to North America are in reality not? Horses, rhinos, elephants and tigers are in fact native to North America and deer, bison, elk and moose are not. You see about a million years ago, horses, rhinos, mammoths, tigers and giant sloths roamed North America. At the same time deer, moose and elk roamed in Asia.
Roughly 30,000 to 20,000 years ago the area that now has open ocean between Alaska and the USSR was dry land. That allowed not only animals to migrate but humans as well. It is not known why horses, rhinos and the other mammals that were once native to North America disappeared. They were on the North American continent until only 8,000 years ago. There have been fossil finds in Canada and Alaska that put the modern horse in North America just 7,000 years ago. By the same token, why did bison disappear from Asia or did those that stayed behind evolve into yaks?
If you are curious you might want to go to your local book store and learn some more about the subject of ancient mammals. There is one called the Beasts of Eden that is quite good.
Since paleontology is an ever changing science, I do not recommend you buy book on sale that is more than a few years old. A case in point – I have several horse books at home and if they were published before 1978 they blatantly state that the horse developed in Asia, which is not true. Horse fossils were discovered in 1928 in Idaho (that were from horses that lived over 1 million years ago) but the general public was not aware of it until the late 1950's. Some authors of those books simply took older books and re-wrote the misinformation, without bothering to do their own research on the matter. In the later books the information about the history of the horse had been corrected.