This is my son's case, he is 11 years old. We thought his status would automatically be extended once one of his parent's H-1B is extended.
-If your son's old H4 does not expire yet, his status is OK even if the parent transferred the h1 to a new employer and get the new H1 I-94. But the son has to extend his H4 by filing I-539 before the old H4 expires using the parent's new H1 approval notice (nothing automatically extended).