A multiharmonic cubic-nonlinear theory of a plasma-beam superheterodyne free-electron laser of the dopplertron type is constructed. A retarded electromagnetic wave propagating in the magnetized plasma-beam system toward the electron beam is used for pumping. The multiharmonic interaction of waves which plays an important role is taken into account. Saturation levels and mechanisms are analyzed. The promising application of such systems for generating high-power electromagnetic radiation in the millimeter wavelength range is demonstrated.