Insect invasions, the establishment and spread of nonnative insects in new regions, can have extensive economic and environmental consequences. Increased global connectivity accelerates rates of introductions, while climate change may decrease the barriers to invader species' spread. We follow an individual-level insect- and arachnid-centered perspective to assess how the process of invasion is influenced by phenotypic heterogeneity associated with dispersal and stress resistance, and their coupling, across the multiple steps of the invasion process. We also provide an overview and synthesis on the importance of environmental filters during the entire invasion process for the facilitation or inhibition of invasive insect population spread. Finally, we highlight important research gaps and the relevance and applicability of ongoing natural range expansions in the context of climate change to gain essential mechanistic insights into insect invasions.