Usage patterns
〈場所〉の跡継ぎになるto become the heir/successor of (a business, family, etc.)
He decided to become the heir to the family business.
跡継ぎがいないto have no heir/successor
This shop has no heir, so they're thinking of closing.
跡継ぎを育てるto train/raise a successor
The president is putting effort into training his successor.
〈人〉を跡継ぎに指名するto name someone as one's heir/successor
My grandfather named his eldest son as his heir.