  Recent progress in constraining the massive accretions (>1:10) experienced by
the Milky Way (MW) and the Andromeda galaxy (M31) offers an opportunity to
understand the dwarf galaxy population of the Local Group. Using zoom-in dark
matter-only simulations of MW-mass haloes and concentrating on subhaloes that
are thought to be capable of hosting dwarf galaxies, we demonstrate that the
infall of a massive progenitor is accompanied with the accretion and
destruction of a large number of subhaloes. Massive accreted progenitors do not
increase the total number of infalling subhaloes onto a MW-mass host, but
instead focus surrounding subhaloes onto the host causing a clustering in the
infall time of subhaloes. This leads to a temporary elevation in the number of
subhaloes as well as changes in their cumulative radial profile within the
virial radius of the host. Surviving associated subhaloes with a massive
progenitor have a large diversity in their orbits. We find that the star
formation quenching times of Local Group dwarf spheroidal galaxies ($10^{5}
\mathrm{M_{\odot}} \lesssim \mathrm{M}_{*} \lesssim 10^{7} \mathrm{M_{\odot}}$)
are clustered around the times of the most massive accretions suffered by the
MW and M31. Our results imply that a) the quenching time of dwarf spheroidals
is a good proxy of their infall time and b) the absence of recently quenched
satellites around M31 suggests that M33 is not on its first infall and was
accreted much earlier.
