The anomalous transport in a one-dimensional (1D), collisionless return-current beam plasma system is studied by means of electrostatic Vlasov and three-fluid simulations in a current-free counterstreaming electron beam situation. Despite of a lack of binary collisions in both approaches, the electron bulk drifts are found to slow down due to collisionless momentum transport and the electrons are heated. It is shown that in the multifluid plasma description, charge separation effects play a major role in this process. A 1D electrostatic Vlasov code simulation is performed with the same macroscopic plasma parameters to investigate the momentum transport mechanism if kinetic effects are taken into account. A comparison of the two different approaches shows that a stronger drift relaxation takes place in the multifluid plasma approach. Also, in both approaches the same drift relaxation level is reached within the same time Δt≈700 ωpe−1. This is due to the common mechanism of anomalous transport—charge separation—in multifluid and kinetic approach during the initial stage of the electron beam relaxation. By carrying out a wave-particle interaction analysis in velocity space, it is shown that, however, the essentially kinetic effects Landau damping and particle trapping stop further current relaxation when the bulk drifts velocity has reached the thermal velocity value. In conclusion, it is argued that in this situation, an additional anomalous transport term is required for a complete multifluid plasma description of the electron beam relaxation.