What phenomenon is responsible for moving sedimentary and metamorphic rock into. the Earth's interior?

The answer is the plate tectonics.

During the rock cycle, the sedimentary rocks return again to the interior earth and become metamorphic rocks by metamorphism, and then the metamorphic rock goes deeper and melt into magma by high temperature.

The plate tectonics movements and the changing in its rate we affect the production and destruction of the sedimentary, metamorphic and igneous rocks.

For example, the metamorphic rocks formed at a position of a compression at convergent plate boundaries, So by increasing the rate of plate tectonic movements the rate of formation of the metamorphic rocks will be increased also.

and sedimentary rocks which meanly moves by weathering and erosion also the plate tectonic affect its movement. as the plate tectonics affect Uplift of the crust areas because the uplift of the crust will occur as a result of the plate tectonics movements.