We analyzed the nonlinear dynamics of pulsed beam self-cleaning in nonlinear tapered Ytterbium doped and Erbium-Ytterbium codoped graded-index multimode optical fibers, with quasi-uniform doping distribution in the core cross-section. By increasing the net gain when operating in active configuration we observed that the output spatial intensity distribution changed from a speckled into a high-quality and bell-shaped beam. By launching pulses in the normal dispersion regime of the taper, from the wider into the smaller core diameter, we generated a supercontinuum emission between 520 nm and 2600 nm. When the laser pulses were launched into the small core side of the tapered fiber or in the Erbium-Ytterbium fiber, self-cleaning was obtained without any self-phase modulation-induced spectral broadening or frequency conversion.