Nonlocal compensation of magnetic damping by spin injection has been theoretically shown to establish dynamic, noncollinear magnetization states that carry spin currents over micrometer distances. Such states can be generically referred to as dissipative exchange flows (DEFs) because spatially diffusing spin currents are established by the mutual exchange torque exerted by neighboring spins. Analytical studies to date have been limited to the weak spin injection assumption whereby the equation of motion for the magnetization is mapped to hydrodynamic equations describing spin flow and then linearized. Here, we analytically and numerically study easy-plane ferromagnetic channels subject to spin injection of arbitrary strength at one extremum under a unified hydrodynamic framework. We find that DEFs generally exhibit a nonlinear profile along the channel accompanied by a nonlinear frequency tunability. At large injection strengths, we fully characterize a magnetization state we call a contact-soliton DEF (CS-DEF) composed of a stationary soliton at the injection site, which smoothly transitions into a DEF and exhibits a negative frequency tunability. The transition between a DEF and a CS-DEF occurs at the maximum precessional frequency and coincides with the Landau criterion: a subsonic to supersonic flow transition. Leveraging the hydraulic-electrical analogy, the current-voltage characteristics of a nonlinear DEF circuit are presented. Micromagnetic simulations of nanowires that include magnetocrystalline anisotropy and nonlocal dipole fields are in qualitative agreement with the analytical results. The magnetization states found here along with their characteristic profile and spectral features provide quantitative guidelines to pursue an experimental demonstration of DEFs in ferromagnetic materials and establish a unified description for long-distance spin transport.