These new mobile app features allow you to move items between your characters and vault.
1) Select a character with the scroller on the menu, then select "Legend."
2)
a) You can take something from the vault by navigating to the far right (the "Vault" tab) within "Legend," selecting an item, hitting "Manage," and then selecting to take it (into the inventory of the character whose Legend you are within).
b) You can put something into the vault by navigating to the "Gear" tab, selecting the item, hitting "Manage," and selecting to place it in the vault. (Note: you can also equip it from this menu, only if you are offline, in orbit, or at the tower.)
When you do this mid-game the item you moved to your character pops up on your screen like a reward drop, it's nearly instantaneous.
Enjoy this new feature! My images are on iPhone, but I would assume it is available on Android as well (and the Bungie.net website).
TL;DR
Legend -> Vault -> Select Item -> Take From Vault
Legend -> Gear -> Select Item -> Move to Vault
edit: I had not seen the Weekly Update when posting this, I just found it when checking my vault. Sorry for re-iterating information!
edit: I already made the post so I might as well pull through with the info. This feature can be used from any point in game, offline to in combat.
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Too many edits, although I think this is more informative than the weekly update. :3