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We present a spatiotemporal picture of human genetic diversity in Anatolia, Iran, Levant, South Caucasus, and the Aegean, a broad region that experienced the earliest Neolithic transition and the emergence of complex hierarchical societies. Combining 35 new ancient shotgun genomes with 382 ancient and 23 present-day published genomes, we found that genetic diversity within each region steadily increased through the Holocene. We further observed that the inferred sources of gene flow shifted in time. In the first half of the Holocene, Southwest Asian and the East Mediterranean populations homogenized among themselves. Starting with the Bronze Age, however, regional populations diverged from each other, most likely driven by gene flow from external sources, which we term “the expanding mobility model.” Interestingly, this increase in inter-regional divergence can be captured by outgroup-f$_3$-based genetic distances, but not by the commonly used F$_{ST}$ statistic, due to the sensitivity of F$_{ST}$, but not outgroup-f$_3$, to within-population diversity. Finally, we report a temporal trend of increasing male bias in admixture events through the Holocene.