This verse depends upon a concept of justice whereby the lineage of a major transgressor would be eliminated. While this is not the Christian concept of not visiting the sins of the fathers upon the children, it is yet part of the Old Testament system of justice. We find in Exodus: Ex. 34:7
“Keeping mercy for thousands, forgiving iniquity and transgression and sin, and that will by no means clear the guilty; visiting the iniquity of the fathers upon the children, and upon the children’s children, unto the third and to the fourth generation.”
In the same concept as Exodus’ “visiting the iniquity of the fathers upon the children, and upon the children’s children…,” Isaiah notes that the descendants of Babylon will also be stripped of their glory. It is not just the king of Babylon who is brought down. It is not just the city of Babylon that is brought down, it is the right of inheritance that is also brought down. Where the world expects that power and wealth will transfer to future generations, Isaiah is clear that the fall of Babylon is so great that it is not temporary, but permanent, and the descendants of Babylon will have no claim to greatness.
2 Ne. 24:22
22 For I will rise up against them, saith the Lord of Hosts, and cut off from Babylon the name, and remnant, and son, and nephew, saith the Lord.
The separation of the descendants of Babylon from their inheritance is directly through the intervention of the Lord of Hosts. When the Lord “cuts off from Babylon the name, and remnant” he is severing the connection between past and future. There will be no inheritance of the former power and glory.