By ‘dropbox’ I assume that you mean the Sorter’s default Inbox slot. Files or clippings dropped into the Inbox in the Sorter are sent to your Global Inbox.
But the Sorter has empty slots. You can drag your various database inboxes (and favorite groups, as well) into empty Sorter slots, and they will be available to receive new content, even if the target database isn’t open at the time.
Tip: To avoid confusion, you may rename the inboxes of your databases before dragging them into a Sorter slot. Example: If you have a database named ‘Harry’, change its Inbox name to ‘Harry Inbox’.