As Cedrad said here, this does need to be done on the character who has the original unlock, in this case that would be your Horde Warrior. Once they have completed the requirements to unlock the other colors though those would be available to use for Transmog by other Warriors on the account (spec restrictions still apply)

You wouldn’t lose the appearances you’d already unlocked those are a part of your Transmog collection, but you wouldn’t be able to unlock the alt colors, as that has to be done on a character with the base appearance.