Experimental observation of the self-induced laser line sweeping (SLLS) in fiber ring lasers is presented. The SLLS with the same gain fiber is studied in Fabry-Perot cavity for comparison. The SLLS effect manifests itself as a laser wavelength drift with speed of the order of nanometer per second from shorter to longer wavelengths across several nanometers and fast backward jump. Recently, the dynamics of the SLLS in a Fabry-Perot cavity fiber laser was qualitatively described by a dynamic grating induced by spatial-hole-burning in the ytterbium doped fiber where the lifetime of the grating was related to the self-sustained relaxation oscillations. In this paper we address possible discrepancies between the published theoretical model and earlier observations of SLLS, particularly in fiber-ring lasers. We show that the qualitative theoretical model developed for explaining SLLS in the Fabry-Perot cavity can be used also to explain the SLLS effect we observed earlier in fiber-ring lasers.