We provide a theoretical description of diffusive charge and spin transport in hybrid devices containing altermagnets. Based on recently derived drift–diffusion equations for coupled charge and spin dynamics and general boundary conditions, our approach provides a unified description of the spin-splitter effect, i.e., the conversion of charge currents into spin currents, and its inverse in terms of experimentally accessible parameters. We analyze, analytically and numerically, the spin-splitter effect, demonstrating that an injected spin accumulation generates a measurable voltage difference across the transverse direction in the altermagnet. Motivated by a recent experiment, we also analyze a nonlocal spin-valve geometry in which an altermagnetic strip injects spin into a diffusive normal metal. We derive the resulting nonlocal voltage detected by a ferromagnetic electrode as a function of the relative orientation of the Néel vector and the ferromagnetic polarization, accounting for the main experimental findings. For this setup, we further address spin precession during diffusive transport by analyzing the spin Hanle effect. Our results provide theoretical explanations and predictions for several altermagnet hybrid structures.