Later on we see how some families delivered plenty of popes like the Borgia and Medici families. If my suspicion is correct then at the time it wasn't even so odd for a pope to have children from before he entered the pontificate functions. With that suspicion in mind it seems logical that there would have been a certain danger of sons of bishops and popes inheriting the positions of their fathers... right?
To the very best of my knowledge no. When you read the books of the Gospel and early church history, you will read almost nothing of the names of the Children of the Apostles. For instance, we know St. Peter was married, because his mother-in-law is mentioned (Matthew 8:14), but we hear nothing about his children. There is some belief that St. Petronilla is his daughter. We also know that St Philip was married and had children.