Ezekiel provides valuable context a few verses later. The latter part of the chapter emphasizes the desire to enable the sinner to repent and, in verses 25-27 indicates that the death being spoken of is of a spiritual nature:
25 Yet ye say, The way of the Lord is not equal. Hear now, O house of
Israel; Is not my way equal? are not your ways unequal?
26 When a righteous man turneth away from his righteousness, and
committeth iniquity, and dieth in them; for his iniquity that he hath
done shall he die.
27 Again, when the wicked man turneth away from his wickedness that he
hath committed, and doeth that which is lawful and right, he shall
save his soul alive.
The back and forth discussion in verse 25 recognizes the tension and asks if what has been stated earlier in the chapter is fair. The Lord points out the eternal nature of the kind of blessings/punishments He has in mind. If the punishment of death indicated by the Lord is physical death, then verse 26 would mean that a dead person would be punished by...dying? How is death a punishment for somebody who is already dead?
The passage makes more sense in light of vs 27, which speaks of saving one's soul. Thus, the punishments noted by the Lord in Ezekiel 18 speak of life or death for the soul--these are predicated upon one's own sins, not the sins of the parents.