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Kinetic pathways of Li-ion diffusion in olivine phosphates are calculated from density functional theory (DFT). Previously reported theoretical diffusion rates for Li ions and vacancies in defect-free crystalline FePO4 and LiFePO4 are six orders of magnitude faster than experimentally measured values. This discrepancy can be resolved by considering the different components of Li kinetics, including diffusion in the bulk, on the surface, in the presence of defects, and in varying local environments. Using DFT+U, we quantify each of these effects and determine that, while bulk diffusion is affected by strain and Li concentration, these are not significant enough to explain the slow diffusion observed in experiment. However, surface diffusion is observed to have have high barriers, which could contribute to slow kinetics in nanostructured cathodes. Anti-site defects also provide a possible explanation for slow diffusion, but only for vacancy diffusion in LiFePO4, which has a barrier of 0.71 eV, compared to 0.29 eV in defect-free channels. In FePO4, a concerted Li-ion diffusion mechanism around the anti-site defect is found to have a low barrier of 0.35 eV, allowing for facile cross-channel diffusion at room temperature. The difference between Li-ion and vacancy diffusion is understood in terms of a favorable coordination between Li ions and localized electrons on Fe centers at the transition states for Li-ion hopping in FePO4. Greater distances between vacancies and holes at the transition states for vacancy diffusion lead to higher diffusion barriers.