We demonstrate that dynamical probes provide direct means of detecting the topological phase transition (TPT) between conventional and topological phases, which would otherwise be difficult to access because of loss or heating processes. We propose to avoid such heating by rapidly quenching in and out of the short-lived topological phase across the transition that supports gapless excitations. Following the quench, the distribution of excitations in the final conventional phase carries signatures of the TPT. We apply this strategy to study the TPT into a Majorana-carrying topological phase predicted in one-dimensional spin-orbit-coupled Fermi gases with attractive interactions. The resulting spin-resolved momentum distribution, computed by self-consistently solving the time-dependent Bogoliubov-de Gennes equations, exhibits Kibble-Zurek scaling and Stuckelberg oscillations characteristic of the TPT. We discuss parameter regimes where the TPT is experimentally accessible.