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?

Lithosphere is the layer of rigid rocks that lies about 80 Km from the asthenosphere which is less denser than the lithosphere and is a fragile viscous layer comprising the upper mantle. The tectonic plates just don't plunge in to greater depths because the asthenosphere is denser than the lithosphere since pressure is high in these areas and the rock does not melt completely in the asthenosphere but the rocks are also not completely solid they are in the molten form depending on the pressure and temperature conditions. Hence, the tectonic plates don't sink in to deeper layer but tend to move because of the lubricating effect provided by the asthenosphere.