Orthodox priests are in some circumstances allowed to be married and have children. Many sons of priests presumably become priests themselves. Have local priests often been succeeded in the job by their own sons? What factors have made this kind of transition more or less likely?
Kislova's "Latin" and "Slavonic" Education in the Primary Classes of Russian Seminaries in the 18th Century cites Matison to say:
The parish was, de facto, a hereditary holding; therefore, the education of future priests became the responsibility of their fathers.
Bremer's Cross and Kremlin: A Brief History of the Orthodox Church in Russia, pp. 78, contributes:
De facto, parishes could be inherited, a practice in Russian (but not Ukranian territories), supported by the hierarchy until the second half of the nineteenth century. It led to the creation of something like a "caste of priests", a social class that frequently had little connection with the rest of society and whose social mobility was extremely limited.... After a reform of seminaries in 1867, sons of priests had an opportunity to choose other professions.
So the answer seems to be, parish inheritance was common at least in Imperial Russia.
To expand on Wtrmute's comment: I don't believe it happens often....but it is possible that it happens. When someone is ordained a Priest, their Bishop decides where they are to serve. If you're talking about the case where a Priest is still serving when their son becomes ordained, then it's more likely the son will be sent elsewhere to serve. If then, say, later the father Priest retires or passes away, the Bishop must decide who takes over, which may or may not end up being the son (though you may think there'd be reason being that the son presumably grew up with that parish, if that son has at that point been serving some other parish a while, why would they be told to leave - "...there is in principle no reason he would prefer the son to any other random priest.").
What can happen is if the timing is just so, where perhaps the son is actually a Deacon when the father retires or passes away, and then the son chooses to become an ordained Priest at the approval of the Bishop to then take over that parish at that time. I have seen this not with father and son but with Archpriest and a long-serving Deacon. When the Archpriest passed away, the Deacon unexpectedly got ordained and took over the parish. But again, that's in both the timing and in the decisions of that Deacon and the Bishop.