The theory of radio windows given in two previous papers for a stratified cold plasma is extended to apply in a warm plasma. It is used to investigate one suggested mechanism for the production of non-thermal continuum radiation in magnetospheric cavities. The source is a plasma wave that enters a region where there is a gradient of electron concentration and there undergoes partial linear mode conversion to give ordinary and extraordinary electromagnetic waves and a reflected plasma wave. This theory is needed particularly for the plasmapause and magnetopause where the concentration gradients may be large. It is therefore necessary to use a full-wave integration of the governing differential equations. These are derived for a warm plasma. When they are integrated, the problem of numerical swamping is severe and is dealt with by a special method. Some typical results are presented and discussed.