Originally Posted by
Qualia
There isn't any detailed explanation, but Blizzard / Muffinus implied that Sargeras and the Burning Legion got them from their Ulduar assaults & invasions. When asked about the matters ("My only point is: last remnants were inside the Keepers. Now, the souls are on Argus. That is a contradiction. They need to explain how"), he answered with "You know how there were demons in Ulduar during Legion pre-launch week 2?", and when the questioner brought up Aman'thul's fragment which should be inside Lei Shen & Wrathion, he said "Did you do invasions?".
With that said, the theories / speculations on WoWpedia aren't a stretch. The only questionable part would be Aman'thul's fragment as Wrathion is nowhere in sight, but maybe either (1) Lei Shen didn't get the entire of Aman'thul's fragment, because he wasn't powerful enough of a vessel or Ra didn't show him the entire thing but just a part of it or (2) The Legion has magic capable of locating and extracting the Pantheon's souls without having to touch the vessel, so as long as they could corner Wrathion in a certain area, they would be able to grab Aman'thul's piece without having to catch Wrathion first.