We study the spatiotemporal dynamics of ultrafast electron spin transport across nanometer-thick copper layers using ultrabroadband terahertz emission spectroscopy. Our analysis of temporal delays, broadening, and attenuation of the spin-current pulse reveals ballisticlike propagation of the pulse peak, approaching the Fermi velocity, and diffusive features including a significant velocity dispersion. A comparison to the frequency-dependent Fick’s law identifies the diffusion-dominated transport regime for distances >2  nm. These findings lay the groundwork for designing future broadband spintronic devices.