Augustine, Enchiridion ad Laurentium:
Chapter 46. It is Probable that Children are Involved in the Guilt Not Only of the First Pair, But of Their Own Immediate Parents.
And it is said, with much appearance of probability, that infants are involved in the guilt of the sins not only of the first pair, but of their own immediate parents. For that divine judgment, I shall visit the iniquities of the fathers upon the children, certainly applies to them before they come under the new covenant by regeneration. And it was this new covenant that was prophesied of, when it was said by Ezekiel, that the sons should not bear the iniquity of the fathers, and that it should no longer be a proverb in Israel, The fathers have eaten sour grapes, and the children's teeth are set on edge. Here lies the necessity that each man should be born again, that he might be freed from the sin in which he was born. For the sins committed afterwards can be cured by penitence, as we see is the case after baptism. And therefore the new birth would not have been appointed only that the first birth was sinful, so sinful that even one who was legitimately born in wedlock says: I was shapen in iniquities, and in sins did my mother conceive me. He did not say in iniquity, or in sin, though he might have said so correctly; but he preferred to say iniquities and sins, because in that one sin which passed upon all men, and which was so great that human nature was by it made subject to inevitable death, many sins, as I showed above, may be discriminated; and further, because there are other sins of the immediate parents, which though they have not the same effect in producing a change of nature, yet subject the children to guilt unless the divine grace and mercy interpose to rescue them.
Chapter 92. The Resurrection of the Lost.
But as for those who, out of the mass of perdition caused by the first man's sin, are not redeemed through the one Mediator between God and man, they too shall rise again, each with his own body, but only to be punished with the devil and his angels…Chapter 93. Both the First and the Second Deaths are the Consequence of Sin. Punishment is Proportioned to Guilt.
And neither the first death, which takes place when the soul is compelled to leave the body, nor the second death, which takes place when the soul is not permitted to leave the suffering body, would have been inflicted on man had no one sinned. And, of course, the mildest punishment of all will fall upon those who have added no actual sin, to the original sin they brought with them; and as for the rest who have added such actual sins, the punishment of each will be the more tolerable in the next world, according as his iniquity has been less in this world.
Source:
https://archive.org/details/saureliiaugusti01augugoog/page/n5/mode/1up