
Your question gives an answer in part - you mention "Registering" the cross bred horses as Sport Horse, or on part-bred registers - the Sport Horse register is specifically fro performance horses, so could, in part be equated to the Agility/Obedience register, on which Cross breds can be registered. Part bred registrations are different. Taking the native breeds, such as New Forests, for instance, the pure bred register, and showing in M & M classes, equates pretty much to the KC pedigree register and the dog showing scene. However, part-breeding with native ponies is how the Show Pony, Working Hunter Pony, etc, is acheived. This is a "Type" not a breed. And there you have the big difference - we do not have a "type" showing classification id dogs, becasue they vary so much in size, shape, coat, etc, etc. However ths "ideal" show horse, outside of the pure breds, are all a pretty similar template, wether it be for a lead rain pony or a show hunter! But in show classes, outsdie of the native and pure breds (mostly in hand, at that) it is the norm to show only geldings and mares - you cannot show a stallion in Ridden show pony, hack or hunter classes. So you need to go to a stallion which will produce to yoru mare. And a mare can produce offspring which will fit several different classes, depending on the stud used - say, you have a 13.2 part-bred arab mare - the other half could be a bit native pony, a bit thoroughbred, a bit unkown. use a 12.2hh Welsh Section B and you could get a lovely SHP type, use a 15 small TB and you can get a 14.2 Show Pony. In dogs there is not direct equal, becasue there is no "ideal" dog shape, just each individual breed standard. And then there is breeding for the various sporting disciplines, show-jumping, dressage, eventing etc. It bear remembering that wiht horses, you have a much more traceable parentage going back to breed beginings, for instance, all TBs go back to 3 Arabians. And another thought, on most of the registries, if you want to go on part-bred or grading register, the horse has to be evaluated (liek the Irish Draqught gradings)
Sorry if this is very fragmented, just wanted to illustrate how very different the horse breeding thing is. Howeve,r one thing remains the same, the novice or uncaring breeder who uses any old stallion on a substandard mare is unlikely to produce anything which will be registered anywhere, although the strangest combinations have managed to produce grade A showjumpers ;)
bye
Gwen