As far as manufacturing, there's a bit of continuation. Studebaker simply went from making wagons to making cars. There are some similarities in appearance, between early cars and buggies, and the same craftsmen who were doing woodwork for coaches , painting, etc. had a decent chance of being hired to build cars.

As far as chauffeurs, There was some concern expressed by new owners of automobiles about the greater independence and power of chauffeurs who drove the car, compared to the ones who'd been employed to drive a carriage. The automobile drivers also tended to function as mechanics ( since early autos required a lot of maintenance) and this and their relative scarcity gave them more leverage over their employers than when they were simply keeping horses.