Question: This is not a training question, but a question related to gait and breeding. If a horse does not gait, but has gaited horses in it's blood line, does it carry genes for gait that will be enhanced by breeding it to a well gaited stallion? Or if it does not gait, does that mean it has no genes for gait?  Or could it be trained to gait?  Most Curly's that gait do a running walk, but a number have been bred to Missouri Foxtrotters and Foxtrot. I have an American Bashkir Curly with a gaited Grandsire on the father's side and gaited Great-grandsire plus others on the mother's side.  I am very interested in breeding her to a Curly Missouri Foxtrotter but don't know if that would ensure a gaited foal?  Also, is it inappropriate to breed laterally gaited horses to diagonally gaited horses?  Thank you. 

Question: This is not a training question, but a question related to gait and breeding. If a horse does not gait, but has gaited horses in it's blood line, does it carry genes for gait that will be enhanced by breeding it to a well gaited stallion?
 
This may or may not work, depending on how the cross is done.  I have seen non-gaited mares bred to gaited stallions that produced gaited offspring -- usually if the stallion was a bit pacey it helped, but it can be done with solid fox trotters too.
 
What gets passed on is some of the conformation traits, and some of the neurological "pathways" that favor gait.  
 
All breeding is a sort of guessing game, seeing what will come out in the foal from each parent ... if you want to do this, be aware that the "gaiting tendency" in the offspring is not guaranteed.
 
However, since it is possible to train a non-gaited horse to fox trot, even if the foal does not naturally appear to choose that gait, you can always teach him later ..
 
Also, is it inappropriate to breed laterally gaited horses to diagonally gaited horses?  Thank you.
 
Depends on what you mean by appropriate -- however, your chance of gait is better with a laterally gaited horse crossed to a diagonally gaited (hard trotting) one than crossing the hard trotter to another diagonally gaited one.   Since just about all gaited horses can do several easy gaits, among them the fox trot which is diagonal, this is probably not
something to worry about excessively.
 
Lee Ziegler