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Over the last several decades, rising pay dispersion between firms accounts for the majority of the dramatic increase in earnings inequality in the United States. This paper shows that a distinct cross-cohort pattern drives this rise: newer cohorts of firms enter more dispersed and stay more dispersed throughout their lives. A similar cohort pattern drives a variety of other closely related facts: increases in worker sorting across firms on the basis of pay, education, and age, and increasing productivity dispersion across firms. We discuss two important implications. First, these cohort patterns suggest a link between changes in firm entry associated with the decline in business dynamism and the rise in earnings inequality. Second, cohort effects imply a slow diffusion of inequality: we expect inequality to continue to rise as older and more equal cohorts of firms are replaced by younger and more unequal cohorts. Back of the envelope calculations suggest that this momentum could be substantial with increases in between-firm inequality in the next two decades almost as large as in the last two.