We investigate the evolution of dust that formed at Population III supernova ( SN) explosions and its processing through the collisions with the reverse shocks resulting from the interaction of the SN ejecta with the ambient medium. In particular, we investigate the transport of the shocked dust within the SNR and its effect on the chemical composition, the size distribution, and the total mass of dust surviving in SNRs. We find that the evolution of the reverse shock, and hence its effect on the processing of the dust, depends on the thickness of the envelope retained by the progenitor star. Furthermore, the transport and survival of the dust grains depend on their initial radius, a(ini), and composition: for Type II SNRs expanding into the ISM with a density of H-n,0 = 1 cm(-3), small grains with a(ini) <= 0.05 mu m are completely destroyed by sputtering in the postshock flow, while grains with (a)ini 0.5-0.2 mu m are trapped into the dense shell behind the forward shock. Very large grains of a(ini) >= 0.2 mu m are ejected into the ISM without decreasing their sizes significantly. We find that the total mass fraction of dust that is destroyed by the reverse shock ranges from 0.2 to 1.0, depending on the energy of the explosion and the density of the ambient ISM. The results of our calculations have significant impact on the abundance pattern of the second-generation stars that form in the dense shell of primordial SNRs.