We establish the connection between the Magellanic Clouds (MCs) and the dwarf galaxy candidates discovered in the Dark Energy Survey (DES) by building a dynamical model of the MC satellite populations, based on an extensive suite of tailor-made numerical simulations. Our model takes into account the response of the Galaxy to the MCs infall, the dynamical friction experienced by the MCs and the disruption of the MC satellites by their hosts. The simulation suite samples over the uncertainties in the MC's proper motions, the masses of the MW and the Clouds themselves, and allows for flexibility in the intrinsic volume density distribution of the MC satellites. As a result, we can accurately reproduce the DES satellites’ observed positions and kinematics. Assuming that Milky Way (MW) dwarfs follow the distribution of sub-haloes in Λ cold dark matter, we further demonstrate that, of 14 observed satellites, the MW halo contributes fewer than 4(8) of these with 68(95) per cent confidence and that 7(12) DES dwarfs have probabilities greater than 0.7(0.5) of belonging to the Large Magellanic Cloud (LMC). Marginalizing over the entire suite, we constrain the number of Magellanic satellites in the range −7 < MV < −1 which exceed the DES surface brightness threshold at ∼70, and the mass of the LMC around 1011 M⊙. The data also strongly support a first-infall scenario for the LMC. Finally, we give predictions for the line-of-sight velocities and the proper motions of the satellites discovered in the vicinity of the LMC.