Is it true that God punishes the oppressor through his children? If so, what is the wisdom behind that, and what is the fault of the children?

There is no reliable documentation regarding a wrongdoer witnessing the punishment of his children, except for a statement attributed to Wahb ibn Munabbih, which mentions that Allah Almighty said: "Indeed, when I am obeyed, I am pleased, and when I am pleased, I bless, and there is no end to My blessing. And when I am disobeyed, I become wrathful, and when I become wrathful, I curse, and indeed the curse from Me reaches the seventh generation of offspring." This statement is from the Isra'iliyyat (Judeo-Christian traditions) which we neither affirm nor deny. And assuming its authenticity, there is no surprise in that, for Allah does what He wills, and He may inflict harm upon the children of the wrongdoer as a punishment for him.