Use former or new location name

I have numerous people who were born in a place with one name, but as time passed it was either renamed or merged into another location. When searching here what do you suggest? Example  born in what was formerly Virginia now West Virginia...another lived in Canajoharie, then Minden, then Danube...although they family did not move. Also  the counties changed, Tryon - Montgomery - Herkimer without the family moving.