Have sons of Orthodox priests often taken over their fathers' ministries? 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.