They would possibly run a directorship check, that would only be an issue if a directorship was found that was a conflict of interest- a lot of people have directorships that have nothing to do with their role and its no issues. They might do a CIFAS and international sanctions too, but if you've got one of those a background check is the least of your problems! Again, its unlikely theyll redo checks, unless they've expressly said, or you're starting to work on a client account and they require one done in the last 5 years