I mean for anyone? What if all horses and their relatives, such as camels, died out before getting to Afroeurasia?
Native Americans wouldn't have any tame horses to wreak havoc on the West US with (the Late 1800s) but then again neither would the United States or its European Colonizers.
It'd probably just slow things down a bit. No one civilization has all the horses, so no one has the advantages. It would just lead to a lot of sore feet and slower armies, communication, etc. until the invention of the automobile and quick, nimble boats.
An interesting cultural side effect is that we no longer measure things in "horsepower".
No horse really changes warfare, which changes a whole bunch of other things in turn. No Mongol/Turkish/Possibly Hunnish hordes, and no rise of feudalism based on a class of cavalry warriors. You may also see more/faster development of maritime technology, as over water trade will become even more important.
I for one welcome our new Siberian overlords.