Once a service worker has activated, any existing Application Cache instances for the client pages that it controls are automatically unassociated. This behavior is described in step 6 of the Activate algorithm in the service worker specification:
For each service worker client client whose creation URL matches
registration’s scope url:
If client is a window client, unassociate client’s responsible
document from its application cache, if it has one.
Else if client is a shared worker client, unassociate client’s global
object from its application cache, if it has one.
If you're not seeing that behavior, and you're sure that the service worker activated and has control over a given client window, then that sounds like a bug in a browsers' implementation. You should follow up with details on reproduction steps on the appropriate browser's bug tracker.