Species invasions are predicted to increase in frequency with global change, but quantitative predictions of how environmental filters and species traits influence the success and consequences of invasions for local communities are lacking. Here we investigate how invaders alter the structure, diversity and stability regime of simple communities across gradients of habitat productivity, temperature, and community size structure. We examine all three-species trophic modules (apparent and exploitative competition, trophic chain and intraguild predation) with empirically derived temperature and body mass scaling of vital rates. We show that the success of an invasion and its effects on community stability and diversity are predictably determined by the effects of environmental factors on each species and the relative strengths of trophic interactions between resident and invading species. We predict that successful invaders include smaller competitors and comparatively small predators, suggesting that species invasions may facilitate the downsizing of food webs under global change.