Thomas Aquinas On Original Sin, Part 2.

From Summa Theologica:

Treatise on Habits in Particular, Question 81, Article 2: Whether also other sins of the first parent or of nearer ancestors are transmitted to their descendants?

If Original Sin is transmitted from Adam to the entire human race, are other sins of ancestors passed to their children? The Scriptures may appear to teach that they are in passages such as Exodus 20:5b, in which God speaks of, "visiting the iniquity of the fathers upon the children to the third and fourth generations of those who hate Me," (NKJV).

Thomas answers the above question in the negative. He points out that,

The reason is that a man begets his like in species but not in individual. Consequently those things that pertain directly to the individual, such as personal actions and matters affecting them, are not transmitted by parents to their children:

Thomas gives the example of one who learns a skill; the skill that is learned is not then passed on to the child. Likewise, a sin that is committed is not passed on to the child. On the other hand, characteristics of a species are passed on to the children. Thomas teaches that the original justice (an upright standing before God) possessed by Adam would have characterized the human species and thus been passed on to his descendants. A chief consequence of Original Sin, however, is that the human race is characterized by sin, and thus a sinful nature is passed on to Adam's descendants. It is interesting to note that in writing on this sinful nature, Thomas defines it as "a proneness to sin."

In regards to Exodus 20:5b, Thomas argues that passages such as Ezekiel 18:4 [Behold, all souls are mine; as the soul of the father, so also the soul of the son is mine: the soul that sinneth, it shall die. (KJV)] into account. Each soul stands accountable before God for his own sins. Thomas understands Ezodus 20:5b to speak of temporal punishments suffered for the sins of parents (as when, to borrow an example he uses elsewhere, the children of a banished criminal suffer by living in a banished condition themselves), and not of spiritual punishments.