If by “building” you mean “training” (in a historical sense) the finished dressage horse should be well capable of performing cross-country and jumping duties before even being started in high school training, which focuses on the pursuit of mobility and control, and does so by harnessing the horse’s ability to collect.
High school equitation emerged, originally, from the requirements of mounted dueling (two (2) adversaries, (2) two horses, (2) swords or lances), where one’s life (and honor) depended upon his ability to manage the horse, and the horses ability to be managed. Thus, these horses were called “managed” or “dressed” or “finished” when their training was complete. The place for exhibiting them – the arena – was called a manege; the process of “dressing” them has come to be called dressage.
Ultimately, mounted dueling (and the unmounted version) was outlawed in Europe, but the art of training and riding the manege horse long continued to be a tradition among noble gentlemen. This was called “indoor” riding, as opposed to “outdoor” or “sport” riding (the latter comprised of flat and cross country racing, as well as hunting – all of which seems to have become popular in England initially, then spreading to the continent to eventually overtake classical manege in popularity, which came to be viewed as a bit high-brow amidst sundry revolutions).
Manege was a fine art – like violin playing – not a ‘sport’ like soccer, or even like modern competitive dressage, where there is a ‘score’ and someone ‘wins’. Much later – once horses were relegated to the role of transportation, after the advent of gunpowder and accurate long-range munitions – the militaries of various countries devised relatively informal contests among officers (and in some cases ‘men’ – enlisted/conscripts) to test the capabilities of remounts (horses used for military service) and horsemanship. These contests challenged endurance (cross-country racing), jumping capability (show jumping), and basic manageability (parade) – the latter emerging to become the basis of modern competitive “dressage”, mostly as a result of the original “3 day Event” becoming too expensive to put on for the Olympics.
In either context, the finished dressage/manege horse should be well capable of performing basic tasks necessary for the military campaign, which certainly include cross-country, jumping, and be manageable on the parade field. This would have been considered a “given” for any horse selected for high school training.
In our savage modern era, the disciplines have become highly specialized. While show-jumping is relatively straightforward in terms of scoring (time is time, faults are evident), competitive dressage is highly subjective, and therefore quite subject to fashion du jour – meaning, for reasons having nothing to do with the precepts of the art’s origins, certain capabilities and manners of riding – and movement in general – come into and out of style periodically – often to the exclusions of others, this being rather the nature of exhibition, or, more specifically, competition.
High school training can only improve the capability and mobility of the jumper, as well as the skills of the rider; jumping capability should be implied when we call a high school horse finished (though many modern dressage show horses would, in reality, be health hazards to their riders beyond the competition rectangle). Competitive dressage horses are increasingly bred for highly stylized, extended trots (in spite of the irrelevance of this movement with respect to classical manege, as evidenced the baroque Spanish-type horses originally bred for it, and still predominately used for the closest modern equivalent, which, contrary to common understanding is not at all ‘competitive dressage’, but rather the mounted bullfighting of Spain and Portugal, which clearly exhibits the rationale for all that is actually defensible with respect to high school training – in my humble opinion).