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We use the APOSTLE $Λ$CDM cosmological hydrodynamical simulations of the Local Group to study the recent accretion of massive satellites into the halo of Milky Way (MW)-sized galaxies. These systems are selected to be close analogues to the Large Magellanic Cloud (LMC), the most massive satellite of the MW. The simulations allow us to address, in a cosmological context, the impact of the Clouds on the MW, including the contribution of Magellanic satellites to the MW satellite population, and the constraints placed on the Galactic potential by the motion of the LMC. We show that LMC-like satellites are twice more common around Local Group-like primaries than around isolated halos of similar mass; these satellites come from large turnaround radii and are on highly eccentric orbits whose velocities at first pericentre are comparable with the primary's escape velocity. This implies $V_{\rm esc}^{\rm MW} (50 $ kpc$)\sim 365$ km/s, a strong constraint on Galactic potential models. LMC analogues contribute about 2 satellites with $M_*>10^5\, M_\odot$, having thus only a mild impact on the luminous satellite population of their hosts. At first pericentre, LMC-associated satellites are close to the LMC in position and velocity, and are distributed along the LMC's orbital plane. Their orbital angular momenta roughly align with the LMC's, but, interestingly, they may appear to "counter-rotate" the MW in some cases. These criteria refine earlier estimates of the LMC association of MW satellites: only the SMC, Hydrus1, Car3, Hor1, Tuc4, Ret2 and Phoenix2 are compatible with all criteria. Carina, Grus2, Hor2 and Fornax are less probable associates given their large LMC relative velocity.