Cool rock material sinks during convection in the mantle because it no longer possesses the heat energy that thrust it upwards away from the intense heat of the outer core.
convection currents in the mantle are the result of the upward motion of molten rock heated deep within the earth. The heat originates in the inner core and permeates through the outer core, enough to keep it in a semi-liquid state.
On contact with the more rocky (less iron) mantle, convection currents are set up in the flowing rock. The rock flows toward the surface, until it cools enough to slow down and stop. Then the flow reverses as the cooler, heavier rock is displaced by the hot flow from below.
Then the cool rock material sinks back towards the inner core, and the process is repeated.
There is more information on the creeping mantle here: