We experimentally investigate characteristics of saturable absorbers (SAs) based on nonlinear Kerr beam cleanup effect (NL-KBC). The SAs are formed by a long graded-index multimode fiber (GRIN MMF) with a short single-mode fiber served as a diaphragm. We studied the evolution of output spectrum and beam profiles from the GRIN MMF in order to investigate the mechanism of these SAs. We further performed saturable absorption measurements to evaluate their modulation depths and saturation intensities. We experimentally observed and first theoretically analyzed the "relaxation oscillation" behavior of the optical transmittance with increasing input intensity. We also studied their nonlinear polarization dynamics and observed the repolarized effect in NL-KBC regime. Our results confirm the optical properties of the SAs based on NL-KBC, and these SAs can find applications in Q-switched and mode-locked lasers.