Doesn't want to work, I've tried all the combinations of moving and turning things off and on. It does 'gather', manually.
*Edit
It seems the problem is only when moving items out of the bank, into inventory. The cookie jars(small and large) will work if you go to the jester, and redeem coins. However, unlike the behavior of other bags, the items will not auto-gather if/when cookie/cake get moved from Shared Bank into character inventory bags. I think it's bugged?