When do items in the "Complete" column on the Kanban board drop off? If they never drop off, it will become an ever-increasing list. If it is based on project status, which status triggers it to drop off?

I understand cards are tasks (stories), but it would be nice to visually see what project they are associated with, when shown on the Kanban Board. Any thoughts on how to accomplish this, without having to click on the caret next to the task name?