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We investigate the K\"ahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat K\"ahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat K\"ahler metrics.