We study the space-time evolution of the fine structure constant, $alpha$, inside evolving spherical overdensities in a lambda-CDM Friedmann universe using the spherical infall model. We show that its value inside virialised regions will be significantly larger than in the low-density background universe. The consideration of the inhomogeneous evolution of the universe is therefore essential for a correct comparison of extragalactic and solar system limits on, and observations of, possible time variation in $alpha$ and other constants. Time variation in $alpha$ in the cosmological background can give rise to no locally observable variations inside virialised overdensities like the one in which we live, explaining the discrepancy between astrophysical and geochemical observations.
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