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 environmental gradients (habitat productivity, temperature) and community size structure. We simulate all three‐species trophic modules (apparent and exploitative competition, trophic chain and intraguild predation). We predict that invasions most often succeed in warm and productive habitats and that successful invaders include smaller competitors, intraguild predators and comparatively small top predators. This suggests that species invasions and global change may facilitate the downsizing of food webs. Furthermore, we show that successful invasions leading to species substitutions rarely alter system stability, while invasions leading to increased diversity can destabilize or stabilize community dynamics depending on the environmental conditions and invader's trophic position.