Nonlinear phenomena are crucial for magnon-based information processing, but the nonlinear interaction between two spin-wave signals requires their spatiotemporal overlap, which can be challenging for directional processing devices. Our study focuses on a gallium-substituted yttrium iron garnet film, which exhibits an exchange-dominated dispersion relation and, thus, provides a particularly broad range of group velocities compared to pure YIG. Using time- and space-resolved Brillouin light scattering spectroscopy, we demonstrate the excitation of time-separated spin-wave pulses at different frequencies from the same source, where the delayed pulse catches up with the previously excited pulse and outruns it due to its higher group velocity. By varying the excitation power of the faster pulse, the outcome can be finely tuned from a linear superposition to a nonlinear interaction of both pulses, resulting in a full attenuation of the slower pulse. Therefore, our findings demonstrate the all-magnonic erasing process of a propagating magnonic signal, which enables the realization of complex temporal logic operations with potential application, e.g., in inhibitory neuromorphic functionalities.