What will eventually happen as the earth's core cools down down over billions of year?

We all die. That's the end of that folks

The liquid outer core would solidify likely at the area of the greatest pressure, at the inner core boundary. This may have two consequences for the core mantle coupling, the angular momentum of the inner core would increase while taking up the angular momentum of the solidifying outer core and perhaps that the reduced convection in the liquid outer core could weaken the magnetic coupling.

Because the increased momentum requires more torque force from the hypothetically weakening magnetic forces in the outer core, the difference in precession between inner core and mantle may increase hypothetically, which could result in greater misalignment of the inner core and mantle spin axes.
This is likely to generate unusual flows in the liquid outer core with all kind of unpredictable effects. One of them might be increased friction and hence conversion of rotational energy into heat, which would potentially heat up the core again??

D-no effect on the rate