The studio did approach Miyazaki's son Goro Miyazaki many times to become a successor to run the studio, but the younger Miyazaki has so far refused all offers, and Hayao Miyazaki also opposes his son becoming the studio's head.

Man, Goro's relationship with his father is an interesting story on it's own. For most part of your childhood your father is barely there, mostly producing movies for other children and when you're grown up you try to catch up to your father in his industry and he thinks that you're fucking worthless.