Alfven's magnetohydrodynamic flux conservation theorem is inapplicable when there are insufficient current-carrying particles to maintain the current that would be required. A generalized conservation theorem, in both integral and differential forms, is obtained here for multi-fluid plasmas with currents having relativistic velocities and with inertial effects included. The fluids are represented as pressure-free ideal gases that do not interact by collisions. For each fluid, hydrodynamic sinks of mass are incorporated to allow for recombination.