We have studied the evolution of material surfaces in strongly time-dependent convection for both Newtonian and non-Newtonian temperature- and depth-dependent rheologies. A spline-characteristic method has been employed. The method of characteristics is second-order in time and fourth-order in space. Our strategy is to employ a very dense grid for the material surface field with about 50 to 100 times more points than for the temperature field grid. We have detected sharp entrainment of the surrounding material into both the ascending and descending flow structures. We have observed the breakoff process involving the descending plume. Mixing takes place very differently for the two rheologies. With time, small vortical features are developed inside the Newtonian plumes, while unmixed islands still persist in the non-Newtonian flow.