  We perform a comprehensive determination of the systemic proper motions of 74
dwarf galaxies and dwarf galaxy candidates in the Local Group based on Gaia
early data release 3. The outputs of the analysis for each galaxy, including
probabilities of membership, will be made publicly available. The analysis is
augmented by a determination of the orbital properties of galaxies within 500
kpc. We adopt the flexible Bayesian methodology presented by McConnachie \&
Venn (2020), which takes into account the location of the stars on the sky, on
the colour-magnitude diagram and on the proper motion plane. We apply some
modifications, in particular to the way the colour-magnitude diagram and
spectroscopic information are factored in, e.g. by including stars in several
evolution phases. The bulk motions are integrated in three gravitational
potentials: two where the Milky Way is treated in isolation and has a mass 0.9
\& 1.6 $\times 10^{12}$M$_{\odot}$ and the time-varying potential by Vasiliev
et al. (2021), which includes the infall of a massive Large Magellanic Cloud
(LMC). We are able to determine bulk proper motions for 73 systems, and we
consider reliable 66 of these measurements. For the first time, systemic
motions are presented for galaxies out to a distance of 1.4 Mpc, in the
NGC~3109 association. The inclusion of the infall of a massive LMC
significantly modifies the orbital trajectories of the objects, with respect to
orbit integration in static Milky Way-only potentials, and leads to 6 galaxies
being likely associated to the LMC, 3 possibly associated and 1 recently
captured object. We discuss the results of the orbit integration in the context
of the relation of the galaxies to the system of Milky Way satellites,
implications for the too-big-to-fail problem, impact on star formation
histories, and tidal disruption.

Astrophysics of Galaxies