It seems USB sticks are not supported.
As mentioned on Xbox forum, on the game case there's a warning:
NOTE: Xbox 360 Hard Drive is required. USB Storage is not supported.
Also, official Xbox 360 game page on the second page of Overview states:
Hard drive required
There's a post on official Bungie forum as well:
You cannot run Destiny off of external flash drives/hard drives -
regardless of storage space, it must be an officially supported hard
drive from the console manufacturer.
So your best bet seems to buy an Xbox HDD. That prerequisite is most likely because of Destiny being online-only game with periodic updates, so it requires both good transfer speed and disk space (update before Dark Below was over 2GB for example).
Destiny lists itself as "requiring an Xbox hard drive" because it requires an internal Xbox hard drive. In this particular case, it turned out that your console was the "arcade" variant, which does not come with an internal drive.
Please keep in mind that future releases will eat up more hard drive space.
Along with the core Destiny game, there are currently three DLCs1;
- The Dark Below
- The House of Wolves
- The Taken King
The Taken King can be considered more of an expansion, and brought a lot of changes to Destiny. These changes effected the vanilla version, too.
Returning players may find that the same occurrence happens with an update, as game updates are frequent, and can often add or change content.
Players that have an account registered prior to September 15, 2015, may be entitle to a free memory upgrade.
By contacting Bungie through their customer support, players may be entitle to receiving a free USB designed for their Xbox 360 or PlayStation 3, with the latest updates from The Taken King.
1At least, as of 6/4/2016.
Best Answer
Normally you can't transfer DLC from one system to another. You'd have to purchase it for both consoles.
In Destiny's case, there was a 'Last Gen to Next Gen' upgrade program for select DLC, but only for a limited time. You probably qualified for this when you bought the first expansion. But it expired on January 15, 2015.
(This is assuming you purchased the DLC individually, instead of in the season pass.)
So unfortunately, if you want the DLC on the 360, you'll have to buy it on the 360.
However, if you purchased the season pass on the 360, then you should try navigating to the DLC page on the Xbox 360 store and see if it is unlocked and just needs to be downloaded.