  The suite of over 60 known planetary debris discs which orbit white dwarfs,
along with detections of multiple minor planets in these systems, motivate
investigations about the migration properties of planetesimals embedded within
the discs. Here, we determine whether any of the migration regimes which are
common in (pre-)main-sequence protoplanetary discs, debris discs and ring
systems could be active and important in white dwarf discs. We investigate both
dust-dominated and gas-dominated regions, and quantitatively demonstrate that
Type I and Type II migration, as well as their particulate disc analogues, are
too slow to be relevant in white dwarf discs. However, we find that the
analogue of Type III migration for particulate discs may be rapid in the dusty
regions of asteroid- or moon-generated ($>10^{18}$ kg) white dwarf discs, where
a planetesimal exterior to its Roche radius may migrate across the entire disc
within its lifetime. This result holds over a wide range of disc boundaries,
both within and exterior to $1R_{\odot}$, and such that the probability of
migration occurring increases with higher disc masses.
