  We present a wavelet-based algorithm to identify dwarf galaxies in the Milky
Way in ${\it Gaia}$ DR2 data. Our algorithm detects overdensities in 4D
position--proper motion space, making it the first search to explicitly use
velocity information to search for dwarf galaxy candidates. We optimize our
algorithm and quantify its performance by searching for mock dwarfs injected
into ${\it Gaia}$ DR2 data and for known Milky Way satellite galaxies.
Comparing our results with previous photometric searches, we find that our
search is sensitive to undiscovered systems at Galactic latitudes~$\lvert
b\rvert>20^{\circ}$ and with half-light radii larger than the 50% detection
efficiency threshold for Pan-STARRS1 (PS1) at (${\it i}$) absolute magnitudes
of =$-7<M_V<-3$ and distances of $32$ kpc $< D < 64$ kpc, and (${\it ii}$)
$M_V< -4$ and $64$ kpc $< D < 128$ kpc. Based on these results, we predict that
our search is expected to discover $5 \pm 2$ new satellite galaxies: four in
the PS1 footprint and one outside the Dark Energy Survey and PS1 footprints. We
apply our algorithm to the ${\it Gaia}$ DR2 dataset and recover $\sim 830$
high-significance candidates, out of which we identify a "gold standard" list
of $\sim 200$ candidates based on cross-matching with potential candidates
identified in a preliminary search using ${\it Gaia}$ EDR3 data. All of our
candidate lists are publicly distributed for future follow-up studies. We show
that improvements in astrometric measurements provided by ${\it Gaia}$ EDR3
increase the sensitivity of this technique; we plan to continue to refine our
candidate list using future data releases.
