It is known that graded-index fibers (GIF) provide efficient Raman conversion of multimode radiation into a Stokes beam with improved beam quality known as Raman beam cleanup effect, whereas in step-index fibers (SIF) this effect is weak. Here we study propagation of low-coherent (~5 nm wide) highly-multimode (M2~30) pump radiation of high power 940-nm laser diodes (LDs) in GIF and SIF of ~100 µm core diameter. We measured pump beam shapes at the output of fibers with different length. It has been shown that the output beam shape is almost independent of the input pump beam shape approaching to rectangular or parabolic profile after propagation in only ten meters of SIF or GIF, respectively. The output beam shape mimics the refractive-index profile in the multimode fiber core due to random mode coupling. This means that Raman gain is nearly the same for all transverse modes in SIF and has maximum gain for the fundamental mode in GIF that is the main reason of strong Raman beam cleanup effect in GIF. We tested Raman lasing in 1-km GIF/SIF with 940-nm LD pumping of up to ~200 W power with a cavity formed by fiber Bragg gratings (FBGs). The output FBG fs-inscribed in GIF/SIF improves the output beam quality due to spatial filtering property, but in GIF the effect is much stronger.