Jumping ability is known to be more hereditary than gaits. There have been quite a few studies on this. I am, however not sure that this translates to the hunter type. It is my understanding that the hunter type is less predictable to breed for hence the reason why many hunter people put greater emphasis on seeing the horse actually jump under saddle vs. gambling with unproven youngsters. There is also a certain type of mindset and movement that is important element for a hunter and this cannot be accurately be determined until they are going under saddle.
We have observed over the years that the movement mechanics STRONGLY resembles the mare's vs the stallion's (watch a lot of videos of babies at mother's side and see it), so we have had an easy time getting the movement we want by starting with mares that move the way we are looking for. A stallion can add elasticity, impulsion, power and scope to the gaits by improving the rear end, but, the mechanics usually mirror the mare. You normally won't find a 10 moving baby out of a 5 mare... except in rare instances where the stallion is just very prepotent for his mechanics (such as Don Alfredo). We always watch lots of offspring videos before picking a stallion to see that if he has knee action, how prepotent is he in passing it. Many times the answer is not very, and why some dressage folks seems so bummed when their baby out of a flat moving mare by, say, Sandro Hit, doesn't have knee. Lots of nice hunters by him (he IS jumper bred after all, hah)... Also have been seeing some six figure hunters by Quaterback.
Jump is supposedly very heretible and we have seen that as well. The stallion can definitely influence in this area - either positively or negatively.
Jumping, hands down. I think this is the accepted thinking by most pro breeders. I was just reading interviews with officials at the Flyinge (Swedish "National Stud") and Celle and both stated that it is far easier to predict jumping ability based on pedigree than movement (dressage).
They both also stated that "prices are better for jumpers, even as foals," and "you might be able to make a dressage horse o/o a jumper-bred horse, but it far more unlikely to produce a jumper o/o a dressage-bred horse."
This was their rationale for breeding more for jumping (Flyinge said they breed 70% jumpers & 30% dressage) and/or establishing a sound jumper breeding program.
I have found (not only in WBs, but other breeds as well) that certain horses are strong in terms of improving movement, while others are not. However predicting the "type" of movement they will produce is not as easy. Some horse will almost always improve movement, but the basic style will stay the same (if that makes sense)
Jumping ability is highly heritable. Horses that have good jumping ability also tend to high atheticism and strong canters. It is why adding jumping lines into a dressage breeding program is helpful at times. Certain breeds have had more focus and success in this area, Holsteiners probably being the poster child. However, it should also be noted that the Trak is also a breed that traditionally has a very good jump (as well as type). It is why I have a foal due May by Buddenbrock Hoping to add that beautiful Trak type and jump (Buddenbrock was Bundeschampionate winner).