If standing on earth, I experience the planet's gravity. Rain drops are accelerated to move toward its centre and reach the crust. These attractions are mutual. This contrasts to e.g., the ISS with reduced / microgravity only. So I started to ponder about Earth's solid core, simplified to float, ball-like, in the outer core known to be a liquid. Equally, I simplify earth to a sphere with a tilted axis of rotation aligned parallel to the straight line from geographic south to north, too.
(modified from this source)
As seen by the tides, and its path around sun, planet earth experiences gravitational attraction with the sun (and its moon), too. If the planet were flexible like a bread dough, instead of a spherical shape, the planet would be constantly kneaded to exhibit a protrusion pointing toward sun, and to lesser degree, moon. Thus I speculate the inner core could equally move slowly / «sink» toward such an external gravitational partner because of its higher density (12.8 to 13 kg/L) compared to the one of the outer core (12.1 kg/L, both taken from here, lacking information about the viscosity of the melt). In the illustration, I marked the position of an excessively displaced core by a blue circle.
Question: Is this line of thought about a «dropping Earth core» correct, is there experimental record for this? In analogy to the ultrasound at the physician, maybe echo waves would travel in shorter time from the mantle to inner core and back if this distance were shorter and thus allow a diagnosis. Given the mass and inertia of the inner core, these imagined movements possibly are small, too.
Moving the core along the line S-N need not be the correct direction of movement to represent here.