Which of the main layers of the earth contains convection currents that slowly move lithosphere plates?

The correct answer is asthenosphere.

The lithosphere is very light and brittle and moves due to convection currents present in the lower layer of the mantle known as the asthenosphere. The Earth's crust is differentiated into small pieces known as plates.

Heat dropping and elevating within the mantle produces convection currents obtained due to radioactive decay in the core. The convection currents move the plates. The movement of the plates and the processes within the Earth is known as plate tectonics.