In the third part of this paper devoted to the problem of the thermal self-focusing of an electromagnetic wave in an underdense plasma, we extend the lineary theory of part 1 to include the effect of ambient drift motion perpendicular to the plane of the axis of the beam and the magnetic field. We show that the amplitude of the density perturbation satisfies a third-order hyperbolic equation, when this effect is included, and we apply Riemann's method to obtain its solution in integral form. This solution shows that a state of stationary self-focusing is established within the beam after a time equal to the transit time for a perturbation to drift across it. The form of the axial growth in the stationary state is derived from an asymptotic evaluation of the integral representation for large axial distances. It is found that the axial growth increases across the beam starting from zero at the edge through which plasma enters the beam and reaching its maximum attainable value at or before the opposite edge, depending on the magnitude of the drift velocity. The maximum growth decreases as the drift velocity increases and only occurs at the rates deduced in parts 1 and 2 when the beam width is several times larger than the scale length which describes the decay of a density perturbation within the beam due to the combined effect of drift longitudinal ambipolar diffusion, recombination and the stabilizing influence of perturbations to the electrical conductivity. The factor by which the amplitude of a convected density perturbation is reduced relative to the value it would have in the absence of drift motion is evaluated and the results are applied to the case of ionospheric modification by a transmission from the proposed solar power satellite systems.