Broadway Director Question

I feel like a theatre noob for asking this, but I was wondering, what is the director's role after a show opens on Broadway? Does he continue to work with the cast, including replacements, for shows that continue running for years? Does Hal Prince still check in on Phantom of the Opera from time to time? If not, what is the title of the person who acts as a "stand in" director for shows that have multi-year runs on Broadway?

How does getting that job work? Does Joe Mantello write down all the minute details of his direction of Wicked, and leave it for whoever is the Resident director?

But a director can continue to be as involved as they choose. Some come back to work with the cast, some come back and give copious notes thru the stage managers.

Some do work with replacement actors. It depends on the ad how, the director...and especially what the director has going on beside that show.