We discuss the dynamical evolution of minor planetary bodies in the outer regions of planetary systems around the progenitors of DZ white dwarfs. We show that during the planetary-nebula phase of these stars, mass loss can lead to the expansion of all planetary bodies. The orbital eccentricity of the minor bodies, as relics of planetesimals, may be largely excited by the perturbation due to both gas drag effects and nearby gas giant planets. Some of these bodies migrate toward the host star, while others are scattered out of the planetary system. The former have modest probability of being captured by the sweeping secular resonances of giant planets, and induced to migrate toward the host star. When they venture close to their host stars, their orbits are tidally circularized so that they form compact disks where they may undergo further collisionally driven evolution. During the subsequent post main sequence evolution of their host stars, this process may provide an avenue which continually channels heavy elements onto the surface of the white dwarfs. We suggest that this scenario provides an explanation for the recently discovered Calcium line variation in G29-38.

Some circumstantial evidence for residual planetesimals is constructed based on the recent discovery of a dusty ring around a young white dwarf at the center of the Helix nebula (Su et al. 2007). This ring extends between about 35 and 150 AU from the nebula center, and have a total mass of about 0.13 M⊕. In this paper we propose that this ring is the by-product of planets and planetesimals' orbital evolution during the epoch when the central star rapidly lost most of its mass. We examine the dynamical evolution of planetary systems similar to the solar system (i.e. with gas giant planets and residual planetesimals) as their host stars evolve off the main sequence. During the process, some planetesimals will be captured by the gas giants into mean motion resonances and their mutual collisions will form a dust ring similar to that observed at the center of the Helix nebula.