By employing element-specific spectroscopy in the ultrafast time scale in Fe1-xNix alloys, we find a composition-dependent effect in the demagnetization that we relate to electron-magnon scattering and changes in the spin-wave stiffness. In all six measured alloys of different composition, the demagnetization of Ni compared to Fe exhibits a delay, an effect which we find is inherent in alloys but not in elemental Fe and Ni. Using a model based on electron-magnon scattering, we extract a spin-wave stiffness from all alloys that show excellent agreement with values obtained from other techniques.