With regard to sons-in-law inheriting a rabbinic position from a father-in-law, Rav Moshe Shternbuch (3:460) quotes the Rama in his teshuvos who says that a son-in-law has the same halacha as a son.
The Avodas HaGershuni argues and says that since the daughter can never inherit the position, the son-in-law has no claim.
Rav Shternbuch holds that the inheriting a position is not based on the real din of “yerushah” like regarding money, but is just a way to honor the previous rov by having his descendants continue to hold his position. Therefore, sons-in-law should have the rights to the position over others if they are qualified, because this will also honor the previous rov.
He distinguishes between a rov who is the boss of a shul and a rov hired by a committee. The prior has a din of a rov regarding this issue, while the latter is simply an employee.
Another distinction Rav Shternbuch makes is between a rov who paskens shailos and is a talmid shochom versus a pulpitrRabbi whose main purpose is to deliver interesting speeches. In the latter case, the congregation has the right to decide whose speeches they would like to hear and is not be bound by inheritance laws.