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We present a kinematic study of six dwarf spheroidal galaxies (dSph) satellites of the Milky Way (MW), namely Carina, Draco, Fornax, Sculptor, Sextans, and Ursa Minor. We combine proper motions (PMs) from the $Gaia$ Data Release 3 (DR3) and line-of-sight velocities ($v_{\mathrm{los}}$) from the literature to derive their 3D internal kinematics and to study the presence of internal velocity gradients. We find velocity gradients along the line-of-sight for Carina, Draco, Fornax, and Ursa Minor, at $\geq 1σ$ level of significance . The value of such gradients appears to be related to the orbital history of the dwarfs, indicating that the interaction with the MW is causing them. Dwarfs that are close to the MW and moving towards their orbits pericentres show, on average, larger velocity gradients. On the other hand, dwarfs that have recently left their orbits pericentres show no significant gradients. Lastly, dwarfs located at large Galactocentric distances show gradients with an intermediate intensity. Our results would indicate that the torque caused by the strong tidal forces exerted by the MW induces a strong velocity gradient when the dwarfs approach their orbits pericentres. During the pericentre passage, the rapid change in the forces direction would disrupt such gradient, which may steadily recover as the galaxies recede. We assess our findings by analysing dwarfs satellites from the TNG50 simulation. We find a significant increase in the intensity of the detected gradients as the satellites approach their pericentre, followed by a sharp drop as they abandon it, supporting our results for the dSphs of the MW.