We report on time- and angle-resolved transverse magneto-optical Kerr effect spectroscopy in the soft x-ray range that, by analysis via polarization-dependent magnetic scattering simulations, allows us to determine the spatiotemporal and element-specific evolution of femtosecond laser-induced spin dynamics in nanostructured magnetic materials. In a ferrimagnetic GdFe thin-film system, we correlate a reshaping spectrum of the magneto-optical Kerr signal to depth-dependent magnetization dynamics and disentangle contributions due to nonequilibrium electron transport and nanoscale heat diffusion on their intrinsic timescales. Our Letter provides a quantitative insight into light-driven spin dynamics occurring at buried interfaces of complex magnetic heterostructures, which can be tailored and functionalized for future optospintronic devices.