When you purchase a DLC, you are purchasing a license to use on that platform it is tied to your platform, not your Bungie account. So you need to own the DLCs on each platform you have linked.
Season passes will share between platforms.
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I dont need dlcs to play with my characters. I know how dlcs work, buddy :) I dont have DLCs on steam and still can play crusible and the free part of the game there ;)