It would just depend on what traits were dominant, what traits were recessive, and what traits were more complex than simple dominant and recessive.
If a homozygous black horse was bred to a homozygous something else horse (say chestnut), the first offspring would all be black. Only in the second cross (hybrid to hybrid) would you get the recessive trait coming out. Then you would have a 3:1 phenotypic ration and a 1:2:1 genotypic ratio.
The first generation would likely be very different from the second. Stuff like the running walk gait may be recessive, or it may involve a lot of different genes. There is a researcher at Cornell that is working on identifying the specific gene right now. I have her email address somewhere. If you want it, PM me.
Cross breeding two purebred animals may result in consistent results. It used to be very typical to breed a black angus cow to a hereford bull (or visa-versa) and you would consistently get a black cow with a white face.
As the cross breeding went past several generations, it all got mixed up.
The National Show Horse is a pretty common breed or actually cross. The Saddlebred and Arab tend to mix up nicely and you often get the Arab style and movement with the Saddlebred size.
I read a post earlier about someone trying to cross Arabs with drafts and getting the Arab size and the draft head. Not good for the home team..........