Interesting topic & one I've heard discussed repeatedly over the years, but before going into it:
Currently our[USA] efforts are represented in the olympic competitions through USA Shooting; it gets zero government support, so any contribution or your membership is most appreciated. We have some wonderful shooters, both male & female and coaching staff. I've spent time on multiple ocassions shooting, learning and laughing w/both Haley Dunn[international skeet] and Bret Erickson [bunker]; we could not be better represented.
www.USASHOOTING.ORG~~~~~~~~~
I'm not aware of trap[called DTL, abroad] ever being shot as a low gun game. In its early iterations it was blood sport. Bill Wise, one of our best loved former members & participants, collected memorabilia from and about 'pig' shoots & shooting, among other things. The earliest ref that he could find to it was a lithograph from the late 1700's that showed the shooter with a pre-mounted flintlock and the bird being released from under a hat. Based on that, early black & white photographs of live bird competitions and the engraved illustrations on old trap guns, I'd say it has always been so .. a mostly pre-mounted gun game or an 'Any Mount' game, if you prefer. One might look at Cyril Adam's style and advise for handling a double [Lock, Stock & Barrel by Adams & his good friend Bob Braden] for a very quick slightly off the shoulder approach & technique in the ring, but Cyril is an exception. For modern clay trap targets, I have never seen nor heard of anyone shooting them low gun in an ATA competition.
On skeet, the game devolved toward the premount due to the nature of tournaments where winners are declared by score and the game evolved into a game in and of itself and it's original purpose as a tool for keeping in tune in the off season was set aside in the quest for perfection as skeet shooting took hold as a sport itself. I know & have seen some wonderful skeet shots who drop the gun off their shoulder before calling for the bird, but they are a dying breed, being bird hunters first and foremost and skeeters second. The game has also gone from pass through shooting with open choked field guns to heavier purpose built tubed guns and sustained leads in order to shoot perfect scores over many targets.
Sporting clays has undergone something similar and so it was predicted that given time, it too would go to the pre-mounted guns and it did just that w/the 'any mount' rule. I heard that being discussed and predicted right after Bob Brister started to advocate the game as sport here. Prior to that the only courses I knew of were some 'Continental' courses otherwise known as 'quail walks'; one at the former Cota de Caza(sp?)in SoCal & one we had in Dallas, later. I'm sure there were others, but I can't speak to them. Those Continental courses were low gun walking affairs with random target releases of singles, pairs, report pairs and sometimes 3 bird flushes that made one pay attention to footing as well as proper mounting.
FITASC remains a low gun discipline within the sporting clays game as does International skeet.
I'm sure others can do a better job of fleshing this out, but think I've covered the basics. Remember that you may shoot clay targets any way you wish from a low gun or off the shoulder or pre-mounted position when you are out w/friends or just practicing proper technique for future time afield. If your goal is to win tournaments then you need to eliminate as many variables as possible and that means pre-mounting more often than not; OTOH good shooting in the field comes from a combination of technique that is inclusive of proper gun mount from somewhere originating off the shoulder.
That's my take onit anyway. Just stay safe & enjoy it, irrespective of your personal bent;-)