We investigate theoretically the motion of chiral Néel domain walls in perpendicularly magnetized systems driven by in-plane strain gradients. We show that such strain drives domain walls efficiently towards increasing tensile (compressive) strain for positive (negative) magnetostrictive materials. During their motion a local damping torque that opposes the precessional torque due to the strain gradient arises. This torque prevents the onset of turbulent dynamics, and steady domain wall motion with constant velocity is asymptotically reached for any arbitrary large strain gradient. Withal, velocities in the range of 500 m/s can be obtained using voltage-induced strain under realistic conditions.