Did you use a hard drive transfer kit when you switched to the new 360? If not, you likely need to re-download the small files for the expansion content licenses. Try this:
https://www.bungie.net/en/Help/Article/12181
Just make sure you are signed in with the same gamertag used to originally purchase the DLC.
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There is an option in setting (I think) that allows you to transfer all your licences to your new Xbox. If you down load all the stuff to your new Xbox without transferring the licenses you will only be able to use them online.
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Thanks!