Kink-like magnetotail flapping oscillations in a Harris-like current sheet with earthward growing normal magnetic field component B_z are studied by means of time-dependent 2D linearized MHD numerical simulations. The dispersion relation and two-dimensional eigenfunctions are obtained. The results are compared with analytical estimates of the double-gradient model, which are found to be reliable for configurations with small B_z up to values ∼ 0.05 of the lobe magnetic field. Coupled with previous results, present simulations confirm that the earthward/tailward growth direction of the B_z component acts as a switch between stable/unstable regimes of the flapping mode, while the mode dispersion curve is the same in both cases. It is confirmed that flapping oscillations may be triggered by a simple Gaussian initial perturbation of the V_z velocity.