Argonne's Model for Universal Solvent Extraction (AMUSE) simulates multistage counter-current solvent extraction processes for species of interest to spent nuclear fuel reprocessing; it is a model of a liquid-liquid extraction unit operation. This work extends the model from its original steady-state implementation to include a time-dependent description of all species of interest. Major components of this extension include a differential mass transfer term, a description of interstage flow rates, and a reaction network for plutonium reduction. The mass transfer term has been formulated using a lumped efficiency term in place of a mass transfer coefficient; the plutonium reduction reactions have been formulated to ensure consistency at all operating conditions. Several nonequilibrium behaviors during simulations have been identified, which allows for improved safety monitoring during process start up and disturbance response.