I don't think six years is enough to clog a heater core. How often do you actually use the heater? Coolant flows through a heater core heating it, and a blower motor than blows air over it and into the cab of the vehicle. Have you had the coolant system serviced at all? That might be the fairly simple fix. After six years, it's probably a good idea anyway.
As for motion, no it shouldn't require motion. It'll heat up faster since the engine will heat faster and the pump will move coolant faster.
If the heater core is truly plugged up and a coolant flush isn't enough, that's a world of fun.