This carrying forward only applies if she has no brothers or other male relatives alive who would inherit the arms on the death of the holder.
His first-born son will inherit the undifferenced (identical) arms on his father's death and pass them on to his descendants.
A similar system applies in Scottish heraldry, though only the oldest daughter is a heraldic heiress.
In this case, her heirs' claim to quarterings through her is dormant, unless and until a petition of arms is granted for the non-armigerous family.
On the death of her husband, her first-born son inherits arms from both parents and quarters them, creating a new coat to pass to his descendants in the usual way.