What process underneath the earth's crust do scientists think causes the movement of earth's plates?

Convection currents.

Explanation:

This is caused by the huge temperature and density differences between plates. The hotter they get, the lighter they are and they rise. On the opposite, colder material is more dense, and therefore, it sinks.

In the deepest parts of the earth's mantle, heat is intense as its close to the core, there, great rock mass is heated, partially melted, and it starts to slowly rise to the surface, this creates an ascendent current. At the same time, near the litosfere, great amounts of cold rock and mass sink back to the mantle, creating a descendent current. This continuous and never ending process is what causes the plate movement.