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.
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