  Observations of the low-mass satellites in the Local Group have shown high
fractions of gas-poor, quiescent galaxies relative to isolated dwarfs, implying
that the host halo environment plays an important role in the quenching of
dwarf galaxies. In this work, we present measurements of the quenched fractions
and quenching timescales of dwarf satellite galaxies in the DC Justice League
suite of 4 high-resolution cosmological zoom-in simulations of Milky Way-mass
halos. We show that these simulations accurately reproduce the satellite
luminosity functions of observed nearby galaxies, as well as the variation in
satellite quenched fractions from $M_* \sim 10^5$ solar masses to $10^{10}$
solar masses. We then trace the histories of satellite galaxies back to $z \sim
15$, and find that many satellites with $M_* \sim 10^{6-8}$ solar masses quench
within 2 Gyr of infall into the host halo, while others in the same mass range
remain star-forming for as long as 5 Gyr. We show that this scatter can be
explained by the satellite's gas mass and the ram pressure it feels at infall.
Finally, we identify a characteristic stellar mass scale of $10^8$ solar masses
above which infalling satellites are largely resistant to rapid environmental
quenching.
