First of all a possible cause for a "wobbling" Earth. I already referred to the Earth as a complex set of different gyroscopes subject to orbital and spinning perturations, causing precession and obliquity cycles. Newton already explained the cause of the precession cycle to be gravity difference on the far side and close side of the equatorial bulge and the moon and the sun, causing a torque force, attempting to erect the Earth spinning axis perpendicular to the plane of the sun / moon orbits. However the spinning causes a precession reaction. This means that the spin axis of the crust-mantle part of the Earth moves around in a cone. But how about the spin axis of the core?
We wonder about the small solid inner core, which shape depends on the pressures and temperature, much more than the centrifugal force. Therefore the solid inner core may not have a equatorial bulge comparable to Earth and even if so, it will still react differently to the Solar and Moon gravitation because the parameters are different. Hence, the solid inner core has a different precession logic independent of the mantle.
The fluid outer core is supposed to stabilize the spin axis of the inner core and acts as a torque converter just like in cars with automatic transmissions. We assume that it's stabilizing capacity is limited and when the size of the solid inner core increases due to cooling, it takes over a tremendous amount of annular momentum, while decreasing the stability capacity of the outer core. As a result, at a certain moment, like, for instance, perhaps 17,5 Ka ago, the spin axis of the solid inner core may break lock with the spin axis of the mantle. This will cause tremendous unstable flows in the outer core, resulting in pressure differences on the core mantle boundary, which ultimately will have dynamic effects on the shape of the Earth.
Next the turbulence will generate additional friction heat, with will cause melting and reduction of the size of the solid inner core again, which may eventually stabilize the complete core again, especially when the inner core has completed it's own independent precession cycle. Then the core would cool again, causing the inner core to grow until again it's spin axis breaks lock and the next cycle starts, say maybe 100 ka years later. Perhaps this whole process started say about one Ma ago, when the solid inner core exceeded stabilitity limits for the first time. Also, the strongly reacting oceans may have caused the Mid Pleistocene Stilostomella extinction event at that time.