Abstract Dwarf galaxies populating the Galactic halo are assumed to host the largest fractions of dark matter, as calculated from their velocity dispersions. Their major axes are preferentially aligned with the Vast Polar Structure (VPOS) that is perpendicular to the Galactic disk, and we find their velocity gradients aligned as well. This finding results in a probability of random occurrence for the VPOS as low as ∼10−5. It suggests that tidal forces exerted by the Milky Way are distorting dwarf galaxies. Here we demonstrate on the basis of the impulse approximation that the Galactic gravitational acceleration induces the dwarf line-of-sight velocity dispersion, which is also evidenced by strong dependences between both quantities. Since this result is valid for any dwarf mass value, it implies that dark matter estimates in Milky Way dwarfs cannot be deduced from the product of their radius to the square of their line-of-sight velocity dispersion. This questions the high dark matter fractions reported for these evanescent systems, and the universally adopted total-to-stellar mass relationship in the dwarf regime. It suggests that many dwarfs are at their first passage and are dissolving into the Galactic halo. This gives rise to a promising method to estimate the Milky Way total mass profile at large distances.