Current theory explains accelerating invasions with increased levels of dispersal as being caused by "spatial selection." Here we argue that another selective force, strong kin competition resulting from high relatedness due to subsequent founder effects at the expanding margin, is of at least comparable importance for dispersal evolution during invasions. We test this hypothesis with individual-based simulations of a spatially structured population invading empty space. To quantify the relative contribution of kin competition to dispersal evolution, we contrast two scenarios, one including kin effects and one excluding them without influencing spatial selection. We find that kin competition is a major determinant for dispersal evolution at invasion fronts, especially under environmental conditions that favor a pronounced kin structure (i.e., small patches, low environmental stochasticity, and high patch isolation). We demonstrate the importance of kin competition and thus biotic influences on dispersal evolution during invasions.