Self-diffusion constants, D, and the atomic-level processes that produce them have been investigated numerically for the binary-mixture Lennard-Jones (BMLJ) model and for liquid silica as described by the Van Beest-Kramer-Van Santen interaction model. The primary conceptual tool for this study is the joint probability distribution for single particles as a function of initial velocity and positional displacement at a given later instant. Self-diffusion constants can be expressed exactly in terms of this probability function. The numerical simulations for the BMLJ case reveal an unusual temperature effect; in contrast to the high-temperature behavior, particles with high initial velocities experience disproportionate retardation in forward displacement. In the silica modeling simulations, diffusive processes have been compared at constant-temperature "isodiffusive" pairs of states, demonstrating a significant role played by the amount of local tetrahedral order that is present in the medium.