Several white dwarfs with atmospheric metal pollution also display evidence for planetesimals orbiting near or within the Roche limit. Such planetesimals likely originated from orbital distances of several AU, suggesting that they underwent a form of high-eccentricity migration. We studied two plausible mechanisms for this migration, (i) tidal friction and (ii) drag forces in a compact accretion disk. We have identified key physical parameters that determine whether the orbit can circularize over an appropriate timescale and constrain these parameters based on the properties of the observed planetesimal systems. If tidal friction is the dominant circularization mechanism, one constrains the internal structure and geophysical properties of the migrating planetesimal. If drag forces are dominant, one instead constrains the characteristic column density of the WD's accretion disk. The dynamics of close-orbiting bodies around polluted WDs provides insight on the properties and processes of extrasolar planetary systems in the terminal stage of their evolution.