The time-dependent behavior of rotating magnetic field (RMF) current drive is investigated using a two-fluid model. The important new factor is the addition of transverse ion mobility in contrast to rigid-ion models of the past. The equations simplify conveniently, allowing the behavior on each surface (r=const) to be isolated, which permits a quadrature solution for the ion fluid rotation. A rapid transient phase leads to quasisteady behavior that evolves on the relatively slow diffusion timescale. The fast transient timescale is set by the ion inertia. Unless there is an ion momentum source to balance the electron drag on the ion fluid, there is no quasisteady current drive effect. Collisions with neutrals offer such a momentum source in some experiments, notably rotamaks and the Star Thrust Experiment. Other sources of ion momentum are essential for RMF current drive in hotter, fusion-relevant plasmas. The properties of the quasisteady state are found, including the self-consistent ion fluid rotation rate and radial electric field, and RMF corrections on the pressure balance.