When magnetic flux moves across layered or granular superconductor structures, the passage of vortices can take place along channels which develop finite voltage, while the rest of the material remains in the zero-voltage state. We study analytically an example of such mixed dynamics: the row-switched (RS) states in underdamped two-dimensional Josephson arrays, driven by a uniform de current under external magnetic field but neglecting self-fields. The governing equations are cast into a compact differential-algebraic system which describes the dynamics of an assembly of Josephson oscillators coupled through the mesh current. We carry out a formal perturbation expansion, and obtain the de and ac spatial distributions of the junction phases and induced circulating currents. We also estimate the interval of the driving current in which a given RS state is stable. All these analytical predictions compare well with our numerics. We then combine these results to deduce the parameter region (in the damping coefficient vs magnetic-field plane) where RS states cannot exist. [S0163-1829(98)11517-5].