  Observations of low-mass satellite galaxies in the nearby Universe point
towards a strong dichotomy in their star-forming properties relative to systems
with similar mass in the field. Specifically, satellite galaxies are
preferentially gas poor and no longer forming stars, while their field
counterparts are largely gas rich and actively forming stars. Much of the
recent work to understand this dichotomy has been statistical in nature,
determining not just that environmental processes are most likely responsible
for quenching these low-mass systems but also that they must operate very
quickly after infall onto the host system, with quenching timescales $\lesssim
2~ {\rm Gyr}$ at ${M}_{\star} \lesssim 10^{8}~{\rm M}_{\odot}$. This work
utilizes the newly-available $Gaia$ DR2 proper motion measurements along with
the Phat ELVIS suite of high-resolution, cosmological, zoom-in simulations to
study low-mass satellite quenching around the Milky Way on an object-by-object
basis. We derive constraints on the infall times for $37$ of the known low-mass
satellite galaxies of the Milky Way, finding that $\gtrsim~70\%$ of the
`classical' satellites of the Milky Way are consistent with the very short
quenching timescales inferred from the total population in previous works. The
remaining classical Milky Way satellites have quenching timescales noticeably
longer, with $\tau_{\rm quench} \sim 6 - 8~{\rm Gyr}$, highlighting how
detailed orbital modeling is likely necessary to understand the specifics of
environmental quenching for individual satellite galaxies. Additionally, we
find that the $6$ ultra-faint dwarf galaxies with publicly available
$HST$-based star-formation histories are all consistent with having their star
formation shut down prior to infall onto the Milky Way -- which, combined with
their very early quenching times, strongly favors quenching driven by
reionization.
