I originally had all the DLC’s when the game wasn’t free. I stopped playing at Forsaken and then I came back during Shadowkeep and during that time I had switched to Xbox from PS4. I enabled cross-save and it transferred my PS4 character to my Xbox Destiny 2, but it never transferred Forsaken and now I’ve got to spend money on a DLC that I’ve already played only to access the Raids and Nightfalls which I find very upsetting and annoying.
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Cross save does not transfer anything even characters. It just allows you to access characters created on one platform's account on another platform. So as there is no transfer your game licences do not transfer to other platforms. If you want the dlc you need to buy it on every platform that you want to play it on.
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Cross Save does not include DLCs, but this is not Bungie’s fault. This is due to licensing between different platforms, which is out of Bungie’s control
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Cross saves does not enable cross dlc. This is tied directly to the platform as it has to do with each platforms licensing policy.