Exodus 20.5. The root for the word translated "punishing" is *pqd*. It means "to attend to (with care); take note; visit (making a visitation, pointing to action that produces a great change in the position of a subordinate either for good or for ill, as in Pharaoh’s servants in Gn. 40: one was restored and the other executed.); appoint; muster or number (of gathering troops or to ascertain the available manpower), reckon." So "punishing in this context is best read as "determine destiny" (to attend to their lives, point to actions, appoint). The sin of the parents here refers to a destiny of calamity or destruction. In other words, the one who sins will face punishment. So when the adult units face demise, so also do the children under them, usually just in the course of normal cause-and-effect. Mistakes by parents live on in their children who are often under their care or even trained in their footsteps. This is not to say the children's fate is set, but only that when a family runs into trouble (especially in their day when family units stayed communally together for generations), that trouble plays itself through many generations. The verse doesn't show that children are punished through no fault of their own, but rather than when the dad messes up, the children often bear some of the consequences (think of gamblers, alcoholics, liars, cheaters, philanderers, and just hateful people). Future generations often suffer the consequences of the mistakes of their predecessors. It's often the case that someone who hates God (Ex. 20.5) teaches their children that same hate (as also true with racism, misogyny, etc.).

Ezekiel 18.20. The theme of Ezekiel 18 is "The One Who Sins Will Face Punishment." It's the same point as Exodus, though expressed differently. Be sure your sins will find you out, and they will have consequences beyond yourself. Certainly the chapter is proverbial, and not a guarantee. In general, all things being equal, the person who lives a good life will live a longer life, and in general, the person who trashes his life won't live as long.

Deut. 24:16 has a similar apparent contradiction. “Fathers shall not be put to death because of their children, nor shall children be put to death because of their fathers. Each one shall be put to death for his own sin."

The reason that this is not a contradiction is because the Exodus passage is descriptive of God's action, while the Deuteronomy passage is prescriptive of how the Israelites should apply the Law and penalties.

I haven't studied the context of this particular passage in Ezekiel, but I would guess that it's a similar situation.

There's also a difference between guilt and punishment. In a class I took, there were several people caught cheating on exams, so the prof. had to restrict when/how people could take the exams. Only a few were guilty, but the whole class was effectively punished. Even future classes will feel the effects of this punishment even though they had no association with the few who were guilty.

Why is God inconsistent with punishment over generations?

Ezekiel 18:20 "The one who sins is the one who will die. The child will not share the guilt of the parent, nor will the parent share the guilt of the child. The righteousness of the righteous will be credited to them, and the wickedness of the wicked will be charged against them."

Exodus 20:5 "You shall not bow down to them or worship them; for I, the Lord your God, am a jealous God, punishing the children for the sin of the parents to the third and fourth generation of those who hate me" What's the reason for this change?