The most obvious difference between solids and liquids is their ability to deform, but that does not mean that only liquids do it. Liquids do not have crystal structures, whereas (some) solids do. Take H2O: water is liquid, but ice is solid. However, glaciers can deform and move. All in the solid state. Metamorphic rocks deform, in the solid state.
Once we accept that solids can deform without being liquid, the ability to sustain convection cells and density-based stratification are not surprising. Fill in a box with gummi bears, then drop a huge block of steel on to it. Eventually the candy will flow upwards while the steel block will sink. All of this will happen in the solid state.