This work presents results of ab-initio simulations of continuous wave transport in disordered absorbing waveguides. Wave interference effects cause deviations from diffusive picture of wave transport and make the diffusion coefficient position- and absorption-dependent. As a consequence, the true limit of a zero diffusion coefficient is never reached in an absorbing random medium of infinite size, instead, the diffusion coefficient saturates at some finite constant value. Transition to this absorption-limited diffusion exhibits a universality which can be captured within the framework of the self-consistent theory (SCT) of localization. The results of this work (i) justify use of SCT in analyses of experiments in localized regime, provided that absorption is not weak; (ii) open the possibility of diffusive description of wave transport in the saturation regime even when localization effects are strong.