Hello, I have recently and luckily found a limited edition at a local Game-stop today and i later wondered the same thing. I see you asked this question in June and I hope this question could be answered by the authority source themselves. I have scoured through most forums and have not found a single legit answer for this question.
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Im no authority, but I believe dlc works the same way as a game. Games are not backwards compatible. So while your character's level progress/stats/story progress will carry over, your dlc probably won't. It could be up to the developer though, they would essentially have to make the dlc free for you on the second platform.