The chiral edge modes of a topological superconductor can transport fermionic quasiparticles with Abelian exchange statistics, but they can also transport non-Abelian anyons: edge vortices bound to a π-phase domain wall that propagates along the boundary. A pair of such edge vortices is injected by the application of an h/2e flux bias over a Josephson junction. Existing descriptions of the injection process rely on the instantaneous scattering approximation of the adiabatic regime [Beenakker, Phys. Rev. Lett. 122, 146803 (2019)10.1103/PhysRevLett.122.146803], where the internal dynamics of the Josephson junction is ignored. Here, we go beyond that approximation in a time-dependent many-body simulation of the injection process, followed by a braiding of mobile edge vortices with a pair of immobile Abrikosov vortices in the bulk of the superconductor. Our simulation sheds light on the properties of the Josephson junction needed for a successful implementation of a flying topological qubit.