Dwarf galaxies provide a unique opportunity for studying the evolution of the Milky Way (MW) and the Local Group as a whole. Analysing the running solar apex based on the kinematics of the MW satellites, we discovered an unexpected behaviour of the dipole term of the radial velocity distribution as a function of the Galactocentric distance. The nearby satellites (<100 kpc) have a bulk motion with an amplitude of 140-230 km s^-1, while the more distant ones show an isotropic distribution of the radial velocities. Such strong solar apex variations cannot be explained by the net rotation of the satellites, as it would require an enormously high rotation rate (≈970 km s^-1). If we exclude the Large and Magellanic Clouds (LMC) and its most closely related members from our sample, this does not suppress the bulk motion of the nearby satellites strongly enough. Nevertheless, we have demonstrated that the observed peculiar kinematics of the MW satellites can be explained by a perturbation caused by the first infall of the LMC. First, we 'undone' the effect of the perturbation by integrating the orbits of the MW satellites backwards (forwards) with (without) massive LMC. It appears that the present-day peculiar enhancement of the solar apex in the inner halo is diminished the most in the case of 2 × 10¹¹ M_⊙ LMC. Next, in self-consistent high-resolution N-body simulations of the MW-LMC interaction, we found that the solar apex shows the observed behaviour only for the halo particles with substantial angular momentum, comparable to that of the MW satellites.