The time evolution of an incompressible non-ideal magnetohydrodynamic (MHD), current-carrying plasma with mass flow is investigated. An approach for the reduction of the nonlinear vector MHD equations to a set of scalar partial differential equations is supposed. Analytical time-dependent solutions of this system are presented. They describe kinetic plasma equilibria both with well-defined nested-in magnetic and velocity surfaces and in the form of vortices. The obtained solutions may be called ‘diffusion-like’, since their temporal structure is very similar to the solutions of the diffusion problem. It is shown that the magnetic field and the velocity have different dumping rates. In the asymptotic limit t→∞, the plasma slowly relaxes towards the hydrostatic equilibrium of gravitating systems.