The nonlinear dynamics of intense truncated Airy beams in Kerr ionizing media are investigated from numerical simulations and experiments. We show numerically that a competition between the linear and nonlinear effects takes place and may be modified by tuning the width of the main lobe of the Airy beam and the size of the truncating diaphragm. Our analysis shows that the acceleration of the Airy peak, an inherent feature of linear Airy beam propagation, is preserved only for powers in the main Airy lobe below a certain threshold. Nonlinear propagation of intense Airy beams with low power in the main lobe is sustained by a continuous energy flux from its neighbors, similarly to the mechanism sustaining nonlinear Bessel beam propagation. Airy beams with higher powers in the main lobe are reshaped into a multifilamentary pattern induced by Kerr and multiphoton nonlinearities. The nucleation of new filaments and their interaction affect the acceleration of the main Airy lobes. We finally show that the size of the truncation constitutes a control parameter for the energy flux that features the Airy beam acceleration. Experiments performed in water corroborate the existence of these two distinct nonlinear propagation regimes.