This may sound like a repeat question, but I've moved from BS -> BL and my Allianz insurance advisor assures me that I only have to change the plates, my driving licence (which states BS as the issuing authority) remains the same and is valid throughout CH.
When I originally changed my Canton Vaud paper licence to a snazzy credit card one in Basel I was charged CHF 75, but now I'm not sure whether I had paid the money for the change from paper to plastic or whether that included a change of issuing authority.