NO, originally one kid retained the original seat assignment, but then the GA was able to get some VDBs so that three of the four would have been able to get onto the flight. All four would have been able to travel if the GA had been willing to give VDBs to the parent and kid who volunteered, but that would have meant that DL would pay one more VDBs than needed and fly with an empty seat (or nonrev).

Also, I'm not sure whether the OP has clearly stated that all four were on the same PNR. That could make a difference in this situation, were seats were available for some but not all of the party.