  From high resolution cosmological simulations of the Local Group in realistic
environment, namely HESTIA simulations, we study the position and kinematic
deviations that may arise between the disc of a Milky Way (or Andromeda)-like
galaxy and its halo. We focus on the 3-dimensional analysis of the centres of
mass (COM). The study presents two parts. We first consider individual
particles to track down the very nature and amplitude of the physical
deviations of the COM with respect to the distance from the disc centre. Dark
matter dominates the behaviour of the COM of all particles at all distances.
But the total COM is also very close to the COM of stars. In the absence of a
significant merger, the velocity offsets are marginal (10 km/s) but the
positional shifts can be important compared to the disc characteristics (>10
kpc). In the event of a massive accretion, discrepancies are of the same order
as the recent finding for the MW under the Magellanic Clouds influence. In a
second part, the accent is put on the study of various populations of subhaloes
and satellites. We show that satellites properly represent the entire subhalo
population. There exists strong mismatch in phase space between the satellites'
COM and the host disc. Moreover, the results are highly inhomogeneous between
the simulations, and thus between the accretion histories. Finally, we point
out that these shifts are mainly due to a few of the most massive objects.
