When the event ends any materials in your bag will remain there but you won't be able to place any more into it.
If you want to withdraw any of the ones stored in there you can at any time and they will move to your inventory, however you won't be able to put them back into the bag.

Note that you can craft from the bag directly, so there is no need to remove the items from the bag as they'll be shared across all of your characters on the account to craft directly from, the daily writs will also take directly from the bag if you run them.

The bank will work in the same way as well, if you've gone over your double cap you will be able to keep the items in there and withdraw them, but you won't be able to place any new items into the bank until it goes back under your allowed limit.