  We investigate the kinematic properties of gas and galaxies in the Local
Group (LG) using high-resolution simulations performed by the {\sc Hestia}
(High-resolution Environmental Simulations of The Immediate Area)
collaboration. Our simulations include the correct cosmography surrounding
LG-like regions consisting of two main spiral galaxies of $\sim
10^{12}$~M$_\odot$, their satellites and minor isolated galaxies, all sharing
the same large-scale motion within a volume of a few Mpc. We characterise the
gas and galaxy kinematics within the simulated LGs, from the perspective of the
Sun, to compare with observed trends from recent HST/COS absorption-line
observations and LG galaxy data. To analyse the velocity pattern of LG gas and
galaxies seen in the observational data, we build sky maps from the local
standard of rest, and the galactic and local group barycentre frames. Our
findings show that the establishment of a radial velocity dipole at low/high
latitudes, near the preferred barycentre direction, is a natural outcome of
simulation kinematics for material {\it outside} the Milky Way virial radius
after removing galaxy rotation when the two main LG galaxies are approaching.
Our results favour a scenario where gas and galaxies stream towards the LG
barycentre producing a velocity dipole resembling observations. While our study
shows in a qualitative way the global matter kinematics in the LG as part of
its on-going assembly, quantitative estimates of gas-flow rates and physical
conditions of the LG gas have to await a more detailed modeling of the
ionization conditions, which will be presented in a follow-up paper.
