One of the surprises for me in working out the answer to this question: Why is the Earth so fat? , is that the core is more elliptical than the surface, the extra ellipticity builds up gradually to about 10% (this is false, I miscalculated the ellipticity, it is actually less elliptical by a factor of 1.47, so 47 percent less ellipticity. This doesn't change the question, except the imagined effects become smaller). As the Earth slows down, the interior has to relax to the new shape, and the inner core is solid. This means that the solid core is the wrong shape for a slower rotating Earth, and it must relieve the gravitational stress by a "core-quake" to move material to the poles from the equator. Unlike the crust, I don't see how it can do this by slow flows, because it isn't floating on anything, it's solid.
I was wondering if anything is known about these core-stresses caused by rotation slowing. Is the core ellipticity wrong for the current rotation of the Earth? By how much? How does the Earth's core relax to the new stable shape when the Earth slows down? What's the time scale? Does this mechanism have any effect on the Earth's magnetic field?
If all of this is unknown, or uncertain, this is a fine answer too.