Well, that was a tricky ride :)
I still have not got any clue how this all worked. Finally I came to the solution of a completion spreadsheet: queue character names and mark them with a “+” if typing “conscript” in the collections tab showed an appearance in every slot.
Mark you, it has to be done and checked character-by-character. For example, earning a cloak for my dwarf shaman does not show for my night elf and draenei hunters. But once the draenei Kalaari earned it, it immediately clicked in for both hunters and – ta-dam – also as a leather cloak for all leather Hordies! This is the most mysterious collection gathering I’ve ever had in WoW.
Anyways:
Alliance Transmog Sets:
Horde Transmog Sets
That done, now we’re going for pet, toy and upgrades. Most weird, most weird event!
Gnomecore 
I took a look at what the armor sets looked like for my Priest. It didn’t jump out to me as a must have. Hopefully Tuesday we will get something new to do.
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With my characters, it seemed like the former Azerite armor slots — head, shoulders, chest — came through to characters of both the same and the opposite faciton rather quickly. Items from the other slots seem to have come through to characters of the same faction before they came through to characters of the opposite faction. But I think if I am patient, everything I’ve bought as an item here and an item there with various characters will eventually show up for all of them. I should have the whole cloth set once the purchases I made today propagate through the system.
I did the whole questchain from start in Orgrimmar to finish in Icecrown this evening and got the “leather” cape. Now I’m contemplating going through with my Shaman and Paladin to get the capes from those sets, too, since capes are not armor-type exclusive. But not tonight. Later in the week, perhaps.
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Hmm, or maybe not… The Wowhead listings say that the cloaks *are* armor-type exclusive.
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Yes, they are :)
Cloaks are even more weird than everything else in this collection: for example, earning a mail Alliance cloak would unlock leather Horde cloak, but not Horde mail. I’ve no idea who was programming this set :)
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