This paper is concerned with one-dimensional and time-dependent multifluid plasma models derived from multifluid MHD equations. In order to reduce the number of equations to be solved, the impurities are described in the framework of the average ion approach without restricting the impurity densities to be small compared with the hydrogen plasma density. Equalizing the plasma temperatures and adopting the condition of quasineutrality, we arrive at a three-fluid description of a current-carrying plasma, and analyse the ability of the self-consistent system of model equations thus obtained to support stationary solutions in a moving frame. This system is reduced to a currentless plasma description assuming at first different flow velocities of the particles and then a currentless, streaming plasma where all particles move with the same velocity. Introducing Lagrangian coordinates and adopting an equation of state, a single reaction–diffusion equation (RDE) for the temperature is obtained. The impurity density, which affects the radiation loss term and the heat conduction coefficient of the RDE, has to be calculated as a function of the temperature by solving additionally a first-order differential equation. This is demonstrated for carbon and high-Z impurities.