Biological invasions can transform our understanding of how the interplay of historical isolation and contemporary (human-aided) dispersal affects the structure of intraspecific diversity in functional traits, and in turn, how changes in functional traits affect other scales of biological organization such as communities and ecosystems. Because biological invasions frequently involve the admixture of previously isolated lineages as a result of human-aided dispersal, studies of invasive populations can reveal how admixture results in novel genotypes and shifts in functional trait variation within populations. Further, because invasive species can be ecosystem engineers within invaded ecosystems, admixture-induced shifts in the functional traits of invaders can affect the composition of native biodiversity and alter the flow of resources through the system. Thus, invasions represent promising yet under-investigated examples of how the effects of short-term evolutionary changes can cascade across biological scales of diversity. Here, we propose a conceptual framework that admixture between divergent source populations during biological invasions can reorganize the genetic variation underlying key functional traits, leading to shifts in the mean and variance of functional traits within invasive populations. Changes in the mean or variance of key traits can initiate new ecological feedback mechanisms that result in a critical transition from a native ecosystem to a novel invasive ecosystem. We illustrate the application of this framework with reference to a well-studied plant model system in invasion biology and show how a combination of quantitative genetic experiments, functional trait studies, whole ecosystem field studies and modeling can be used to explore the dynamics predicted to trigger these critical transitions.