I think you have to purchase the DLC again on 360 to have access to that content on the 360. There was some deal to have Dark Below access purchased on the 360 transferred to One, but they didn't offer it the other way around.
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Yeah this happened to me
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So did you do the same whereas you had xbox one version first but also tried to play the x360 version to not have the dlc available to you?
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I bought the game on 360 and that made it on for both, I bought the expansion on the xbox one first and ended up having to buy it again on 360