If the brittle lithosphere consists of individual solid plates of rock and the asthenosphere is a crystal mush that flows very slowly, why don't the tectonic plates just sink to great depths and melt?

Mantle convection is the process that drives plate tectonics. the theory of plate tectonics states that the crust is divided into plates that float on the semi molten mantle below. the circular motion of the magma as it slowly rises and falls or move though convection cells drags the plates along causing them to move. the heat source in the lamp would represent the earth's core. the core is the source of energy that drives convection in the mantle. the lava represents molten magma under the earth's surface. if the lava lamp had floating pieces they would move as the circular motion of the lava in the lamp as it rises and falls would move them along the surface. these floating pieces would represent pieces of the earth's crust.