[2021] MOVED: is there any way to move items quickly from your inventory to a storage box?

There are buttons built into the UI for this. One button pushes all items that have existing stacks in the storage, and the other button move all items to the storage. You can do the same going from storage to backpack as well, which is useful for a quick restock of ammo or food.

You can also shift-click instead of using the drag-and-drop method.