What is the fundamental process underlying the movement of the plates?

Heat and gravity are fundamental to the process

The energy source for plate tectonics is Earth's internal heat while the forces moving the plates are the "ridge push" and "slab pull" gravity forces. It was once thought that mantle convection could drive plate motions

Tectonic plates

Explanation:

Tectonic plates are able to move because the Earth's lithosphere has greater mechanical strength than the underlying asthenosphere. Lateral density variations in the mantle result in convection; that is, the slow creeping motion of Earth's solid mantle.