  We explore the quenching of low-mass galaxies (10^4 < Mstar < 10^8 Msun) as a
function of lookback time using the star formation histories (SFHs) of 38 Local
Group dwarf galaxies. The SFHs were derived from analyzing color-magnitude
diagrams of resolved stellar populations in archival Hubble Space
Telescope/Wide Field Planetary Camera 2 imaging. We find: (1) Lower mass
galaxies quench earlier than higher mass galaxies; (2) Inside of virial radius
there is no correlation between a satellite's current proximity to a massive
host and its quenching epoch; (3) There are hints of systematic differences in
quenching times of M31 and Milky Way (MW) satellites, although the sample
sample size and uncertainties in the SFHs of M31 dwarfs prohibit definitive
conclusions. Combined with literature results, we qualitatively consider the
redshift evolution (z=0-1) of the quenched galaxy fraction over ~7 dex in
stellar mass (10^4 < Mstar < 10^11.5 Msun). The quenched fraction of all
galaxies generally increases toward the present, with both the lowest and
highest mass systems exhibiting the largest quenched fractions at all
redshifts. In contrast, galaxies between Mstar ~ 10^8-10^10 Msun have the
lowest quenched fractions. We suggest that such intermediate-mass galaxies are
the least efficient at quenching. Finally, we compare our quenching times with
predictions for infall times of low-mass galaxies associated with the MW. We
find that some of the lowest-mass satellites (e.g., CVn II, Leo IV) may have
been quenched before infall while higher mass satellites (e.g., Leo I, Fornax)
typically quench ~1-4 Gyr after infall.
