We investigate the kinematic properties of gas and galaxies in the Local Group (LG) using high-resolution simulations performed by the 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 ~1012 M⊙, their satellites and minor isolated galaxies, all sharing the same large-scale motion within a volume of a few Mpc. We characterize 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 LG 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 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 ongoing assembly, quantitative estimates of gas-flow rates and physical conditions of the LG gas have to await a more detailed modelling of the ionization conditions, which will be presented in a follow-up paper.