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Modern psychology has largely validated the idea that many behaviors are passed on from generation to generation. Medicine has always known that this is the case with many bodily conditions. This involves both good and bad things. Thus, it is clear that Divine Providence so governs human life as to include also the passing of both good and bad things from one generation to the next. Indeed, a foundational and essential Christian doctrine, the doctrine of original sin is such a teaching. So there is nothing that would distinguish the Old and the New Testaments in this regard.
Simply, under the New Testament the means of overcoming an inherited evil or of making the best use of an inherited good would be more powerful and effective. Perhaps it is for this reason that under the New Testament we do not emphasize the generational aspect of human qualities as we might have without the helps and remedies we possess.
One thing is certain: personal moral guilt cannot be inherited, even if some of the effects of personal moral guilt can be. You rightly point out that the text of John 9: 1-3 does not rule out the possibility of a punishment being passed on from one generation to the next. Indeed, Our Lord’s teaching that it was for the glory of God that the man was born blind would have to be true, even if one reason for his blindness had been the punishment or correction of his parents. One can well imagine, for example, a child born with some defect because of his mother’s unhealthy or immoral behavior during her pregnancy overcoming this defect with courage so that God’s work could be made manifest in him through Christ.