Controlled deposition of high laser-power density at remote distances still remains a challenge. Previous experimental works using terawatt peak-power laser systems to initiate filamentation at distances of hundreds or even thousands of meters in the atmosphere have documented limited control on laser energy deposition due to the effects of diffraction and turbulence. In this work, we demonstrate a promising scenario for the projection of high power densities at kilometric distances in air, which requires multiple-gigawatt laser pulses reshaped into ring-Airy beams. We show that a power distribution over a relatively large primary ring accompanied by the inward power flux characterizing a ring-Airy beam limits the occurrence of nonlinear effects to the vicinity of the focal point. Close to focus, self-focusing sets in, which accelerates the convergence process of the beam. We quantify the influence of self-focusing on the focal shift and propose an empirical law that fits our numerical simulation results for the position of the nonlinear focus as a function of the input power and the apodization factor of the ring-Airy beam. We show that once the intensity of the beam exceeds the ionization threshold of air, a short filament is generated whose power content rather accurately corresponds to the critical power for self-focusing, independent of the input parameters.