The common-cause hypothesis says that factors regulating the sedimentary record also exert macroevolutionary controls on speciation, extinction, and biodiversity. I show through computational modeling that common cause factors can, in principle, also control microevolutionary processes of trait evolution. Using Bermuda and its endemic land snailPoecilozonites, I show that the glacial-interglacial sea level cycles that toggle local sedimentation between slow pedogenesis and rapid eolian accumulation could also toggle evolution rates between long slow phases associated with large geographic ranges and short rapid phases associated with small, fragmented ranges and “genetic surfing” events. Patterns produced by this spatially driven process are similar to the punctuated equilibria patterns that Gould inferred from the fossil record of Bermuda, but without speciation or true stasis. Rather, the dynamics of this modeled system mimic a two-rate Brownian motion process (even though the rate parameter is technically constant) in which the contrast in rate and duration of the phases makes the slower one appear static. The link between sedimentation and microevolution in this model is based on a sediment-starved island system, but the principles may apply to any system where physical processes jointly control the areal extents of sedimentary regimes and species distributions.Non-Technical SummaryThe history of life is known from the fossils preserved in the geological record. The common-cause hypothesis suggests that processes like mountain building and sea-level change can affect both the structure of the geological record and species diversity. Using the snails of Bermuda as an example, this paper develops a computational model to show that sea level cycles could affect morphological evolution within species, not just species diversity. As sea level rose on Bermuda, the available snail habitat would have become smaller and more fragmented, which would be expected to drive rapid bursts of genetic drift (the random component of evolutionary change). When sea level fell, the snails’ habitat would have expanded and coalesced, resulting in slower rates of evolution because the total population size would increase. This process would produce an uneven cycle of rapid and slow evolution similar to what paleontologist Steven Gould observed in the fossil snails of Bermuda and led him to propose the theory of punctuated equilibria. While these simulations are focused specifically on an island system, the principles are applicable to other situations suggesting that geological and evolutionary processes may be linked in more ways than previously understood.