We analyze the nonlinear evolution of differential streaming between core protons and alpha particles (or any secondary minor protons) based on the analysis of fully nonlinear multifluid MHD equations. It is shown that such an ion beam-plasma system possesses an equilibrium state with a remnant of differential streaming. The properties of this state are studied. The system through the action of nonlinear hydromagnetic waves can attain an equilibrium state which is characterized by a gyrating motion of both ion populations with a phase shift of π. The waves which bring about a such transition are either the Alfven or magnetosonic modes, depending upon the beam speed. It is shown that the equilibrium differential speed between the alphas and protons Vαp lies between 0.8 and 1.5 VA but only in a certain regime of injection speeds of the alphas. A similar picture obtains for proton/proton configurations in which the number density of the secondary protons imposes a constraint on the injection speeds, 1.5 ≥ Vp,p/VA ≥ 0.3 (VA is a local Alfven speed). Nonlinear Alfven waves bring about a transition to equilibrium states with much higher differential velocities uαp(upp), proportional to the injection speeds, than magnetosonic waves.