At the moment, if you complete a World of Warcraft quest and earn rewards with some gear, only items that your character can use will be added to your cosmetic wardrobe, where all your Cosmetic Wardrobe Transmog options are stored. This means that if you are a leather user, the alt wearing your plate won't get any goodies, except maybe a cloak. This tyrannical system is nearing an end with a war within. 

The system was teased when Blizzard announced Warbands. But now that players are testing the war, we're actually getting to see how it works. 

In the upcoming expansion, completing a quest will unlock all transmog options in your account, regardless of weapon or armor type (Cheers, Wowhead). So if you receive a leather boot as a quest reward, you can also unlock the equivalent of mail, plate, and cloth. Similarly, Evokers can't use shields, but evokers can collect the appearance of shields.

The caveat is that only classes that can use gear will now be able to use transmogs, which can only be applied to the correct armor and weapon types. So, you can't show your Evoker like they're wearing plate armor. 

In addition to this more equal approach to transmogs, the appearance menu has also been tweaked to allow you to see the entire collection, regardless of the class you're playing. In a live game, you can only see the transmog option for the character you are currently playing, but in War, you can see everything by selecting a class from the drop-down menu. 

I get out of my way to make sure my character has the best drip and lately I make my Death Knight look perfect

One question remains, however: "Can I still run through quests and achievements again?" I have my doubts, to be honest, but I have crossed my fingers.