SQL 2012 Availability Group issue - The core cluster resources do not move when we initiate AG fail-over
There is no reason to move the cluster core resources. They are NOT part of the availability group and function entirely on their own. There is no need to even have these on the same server as any of your AG resources.

The core cluster group provides the administrative point for the cluster which is completely independent from any other resources on the cluster. Leave it migrate on its' own.