What will happen to the convection currents in the mantle if the Earth's interior eventually cools down?
As the magma cools and solidifies, the convection currents will stop and the Earth will become geologically dead.
Convection currents within the Earth's mantle are caused by hot material rising upwards, cooling, then dropping back toward the core. These currents are thought to be the driving force for tectonic plate activity in the crust. The moving magma in the mantle carries the plates floating on top of it.
As a result of convection, the Earth's crust is constantly being created and destroyed. The average age of the Earth's surface is 2-2.5 billion years, which is about half the theoretical age of the Earth!
If the interior of the Earth were to cool enough for convection currents to stop, then the motion of the plates would cease, and the Earth would become geologically dead. Astronomers believe that this happened to Mars long ago.
Many of Mars's surface features suggest that at one point Mars had tectonic plate activity. Today, however, Mars is a static world and the only geologic activity is in the form of erosion and bombardment from space.