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Predicting how quickly populations expand their range and whether they will retain genetic diversity when they are introduced to new regions or track environmental conditions suited to their survival is an important applied and theoretical challenge. The literature suggests that long‐distance dispersal, landscape heterogeneity and the evolution of dispersal influence populations' expansion rates and genetic diversity. We used individual‐based spatially explicit simulations to examine these relationships for Tench (Tinca tinca), an invasive fish expanding its geographical range in eastern North America since the 1990s. Simulated populations varied greatly in expansion rates (1.1–28.6 patches year−1) and genetic diversity metrics, including changes in observed heterozygosity (−19 to +0.8%) and effective number of alleles (−0.32 to −0.01). Populations with greater dispersal distances expanded faster than those with smaller dispersal distances but exhibited considerable variation in expansion rate among local populations, implying less predictable expansions. However, they tended to retain genetic diversity as they expanded, suggesting more predictable evolutionary trajectories. In contrast, populations with smaller dispersal distances spread predictably more slowly but exhibited more variability among local populations in genetic diversity losses. Consistent with empirical data, populations spreading in a longer, narrower dispersal corridor lost more neutral genetic variation to the stochastic fixation of alleles. Given the unprecedented pace of anthropogenic environmental change and the increasing need to manage range‐expanding populations, our results have conservation ramifications as they imply that the evolutionary trajectories of populations characterised by shorter dispersal distances spreading in narrower landscapes are more variable and, therefore, less predictable. [ABSTRACT FROM AUTHOR]