I am not aware of such a map - which would be difficult as several families migrated several times, These locations would then have to be date related and 'we' do not have that much information.
Our paths cross with the parents of Thomas K King. His parents were Thomas (Apr 10, 1768 Loudoun Co, VA - Jun 5, 1836 Buford, Ohio Co, KY) and Elizabeth (ca 1768 VA - 1803 Gallatin Co, KY) (Cotton) King. This Thomas is occasionally erroneously reported as Thomas K. King. To my knowledge, he did not have a middle name/initial.