Guilt across generations
Visiting the fathers' iniquity on the children beside Ezekiel's 'the soul who sins shall die' (also Deuteronomy 24:16; Jeremiah 31:29-30) — corporate and individual responsibility.
Exodus 20:5-6
5you shall not bow yourself down to them, nor serve them, for I, Yahweh your God, am a jealous God, visiting the iniquity of the fathers on the children, on the third and on the fourth generation of those who hate me,
6and showing loving kindness to thousands of those who love me and keep my commandments.
Ezekiel 18:19-20
19“Yet you say, ‘Why doesn’t the son bear the iniquity of the father?’ When the son has done that which is lawful and right, and has kept all my statutes, and has done them, he will surely live.
20The soul who sins, he shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son. The righteousness of the righteous shall be on him, and the wickedness of the wicked shall be on him.
What each tradition says
Rabbinic / Midrashic
Rashi reads “of them that hate me” in the same sense as the Targum: the verse concerns those who retain in their hands, or follow the example of, their ancestors’ evil doings. He cites Sanhedrin 27b for this explanation.
Rashi (Rabbi Shlomo Yitzhaki, 1040-1105), Pentateuch with Rashi's Commentary (trans. M. Rosenbaum and A. M. Silbermann, 1929-1934; Sefaria text) — comment on Exodus 20:5 ('of them that hate me')
Patristic
Chrysostom answers the objection about God visiting the parents’ sins on the children to the third and fourth generation by saying that this assertion is not universal. It was spoken with reference to certain people who came out of Egypt.
John Chrysostom (c. 349-407), Homilies on the Gospel of St. John, in Nicene and Post-Nicene Fathers, ser. 1, vol. 14 (The Oxford translation, ed. Philip Schaff; Buffalo: Christian Literature Publishing Co., 1889) — Homily LVI, on John 9:2-3
Catholic
Aquinas says the fathers' sins are punished in their children because children are more likely to sin by being raised among their parents' crimes—getting used to them and imitating their parents' example, as if following their authority. They deserve heavier punishment if, seeing their parents punished, they do not change their ways.
Thomas Aquinas (c. 1225-1274), Summa Theologica, Part I-II, q. 87 (trans. Fathers of the English Dominican Province, 1920; New Advent text) — I-II, q. 87, a. 8, reply to objection 1 ('Whether anyone is punished for another's sin')
Orthodox
Cyril of Alexandria says penalty is for those detected in crime. It was ordained, he says, that only those obnoxious to the law should be punished.
Cyril of Alexandria (c. 376-444), Commentary on the Gospel according to S. John, vol. II (Library of the Fathers; Oxford: James Parker, 1874), Book VI — on John 9:2-3 (excursus on Exod. 34:7 / Ezek. 18)
Reformed / Protestant
Calvin says the good, even born of wicked parents, should receive righteousness’s reward as certainly as if from heaven or a family without crime. Since God does not punish them for their fathers’ crimes, the Israelites’ taunt that their teeth were set on edge because their fathers ate sour grapes was foolish and impious.
John Calvin (1509-1564), Commentaries on the First Twenty Chapters of the Book of the Prophet Ezekiel, vol. 2 (trans. Thomas Meyers; Edinburgh: Calvin Translation Society; CCEL text) — Commentary on Ezekiel 18:14-17
Evangelical
John Wesley’s position is that God is not unrighteous if parents die in their iniquity and their children walk in their steps. When God comes in judgment to reckon with those children, he brings their fathers’ idolatries into account.
John Wesley (1703-1791), Explanatory Notes upon the Old Testament (1765; CCEL's 'Wesley's Notes on the Bible') — note on Exodus 20:5