Based on the flat-topped beams model proposed by Li and the propagation theory of partially coherent light, the spectrum analytical expression of partially coherent flat-topped beams propagating in the linear gain or loss media was derived and the spectral properties were studied quantitatively by numerical calculation. The variation characteristics of the on-axis spectrum in near- and far-field regions have been discussed in details. The effects of the property of the linear gain or loss media and some parameters of the light source, such as the central frequency, spectrum width and spatial coherence parameter on the spectral shift have also been discussed. It can be shown that the light source parameters have greater effects on the value of the frequency shift than on the total changing trend of the spectrum along the propagation distance, whereas the latter is determined by the beam order of flat-topped beam.