The Rambam rules that not only the kingship, but rather every position of authority and all appointees of Israel are inherited to a son and to a son’s son, forever. The Ginas Veradim quotes an anecdote of how the rabbinate in Tzefas was passed on by inheritance to a son who was under bar mitzvah when Rav Shlomo Alkabetz ordered the community to wait for the young boy to come of age, and when the son turned thirteen he took over his father’s position! The Shulchan Aruch bases a ruling on the same halachah, this time in connection with the local cantor, who retains the right to bring in his son as an assistant and groom him for taking over his position (Orach Chaim 53:25).
In Yoreh Deah (245:22), the Rema states that somebody who serves as the rabbi of a city cannot be ousted from his standing even if somebody greater than him comes to town. Even his son and his son’s son, forever, take precedence over others. In Orach Chaim 13 the Chasam Sofer writes that he upheld the ruling of Rema that the son inherits his father’s rabbinical position and another Chasam Sofer (Choshen Mishpat 21) also implies this. The Kesav Sofer (the Chasam Sofer’s son, Yoreh Deah 123) confirms that his father implemented the Rema’s ruling in all Hungarian rabbinates that the son inherits the rabbanus. The Mishna Berura (53:83) cites both opinions.