The influences of the depth-dependent thermal expansivity and viscosity on mantle flows have been investigated with an axisymmetric spherical-shell model and a comparison has been made with Cartesian box results. This comparison between the Cartesian and spherical-shell geometries shows that there are present in both configurations large thermal plumes, while the downwellings are stronger in the Cartesian geometry. Spherical models with a small core, as perhaps in the case of Mars, produce huge megaplumes with large heads, which can extend several tens of degrees on the planetary surface. We have also investigated the influence of the Rayleigh number, internal heating and depth-dependent properties on the time-dependent phase-space trajectories of the dynamically induced moments of inertia and the surface Nusselt number. Large, homoclinic-like, excursions in the phase-space trajectories can occur occasionally in the depth-dependent models and are due to the time-dependent nature of the plume dynamics. The time-scales associated with changes of the surface Nusselt number are faster than those associated with variations in the moment of inertia for earth-like Raleigh numbers. There are substantial temporal variations in the moment of inertia due to the plume-plume collisional dynamics. In the case of Earth, the magnitudes of perturbed moment of inertia may reach as large as 10-5 of the principal moments of inertia. Such a possibility points to a fundamentally important role played by the lower mantle dynamics in polar wander over geological time scales.