We performed global MHD simulations to investigate the magnetotail response to the solar wind directional changes (V(z)-variations). These changes, although small, cause significant variations of the neutral sheet shape and location even in the near and middle tail regions. They display a complicated temporal response, in which similar to 60 to 80% of the final shift of the neutral sheet in Z direction occurs within first 10-15 min (less for faster solar wind), whereas a much longer time (exceeding half hour) is required to reach a new equilibrium. The asymptotic equilibrium shape of the simulated neutral sheet is consistent with predictions of Tsyganenko-Fairfield (2004) empirical model. To visualize a physical origin of the north-south tail motion we compared the values of the total pressure in the northern and southern tail lobes and found a considerable difference (10-15% for only 6 degrees change of the solar wind direction used in the simulation). That difference builds up during the passage of the solar wind directional discontinuity and is responsible for the vertical shift of the neutral sheet, although some pressure difference remains in the near tail even near the new equilibrium. Surprisingly, at a given tailward distance, the response was found to be first initiated in the tail center (the 'leader effect'), rather than near the flanks, which can be explained by the wave propagation in the tail, and which may have interesting implications for the substorm triggering studies. The present results have serious implications for the data-based modeling, as they place constraints on the accuracy of tail magnetic configurations to be derived for specific events using data of multi-spacecraft missions, e.g. such as THEMIS.