Unsteady nonlinear magnetization dynamics are studied in an easy-plane ferromagnetic channel subject to spin injection at one edge. The Landau-Lifshitz equation is known to support steady-state solutions, termed dissipative exchange flows (DEFs) or spin superfluids. In this work, by means of numerical simulations and theoretical analysis, we provide a full description of the injection-induced, large-amplitude, nonlinear magnetization dynamics up to the steady state. The dynamics prior to reaching steady state are driven by spin injection, a perpendicular applied magnetic field, the exchange interaction, and local demagnetizing fields. We show that the dynamics result in well-defined profiles in the form of rarefaction waves (RWs), dispersive shock waves (DSWs), and solitons. The realization of these coherent structures depends on the interplay between the spin injection strength and the applied magnetic field. A soliton at the injection boundary, signaling the onset of the magnetic 'supersonic' condition, rapidly develops and persists in the steady-state configuration of a contact soliton DEF. We also demonstrate the existence of sustained soliton-train dynamics in long time that can only arise in a nonzero applied magnetic field scenario. The dynamical evolution of spin-injection-induced magnetization dynamics presented here may help guide observations in long-distance spin transport experiments.