Many facets of ecological theory rely on the analysis of invasion processes, and general approaches exist to understand the early stages of an invasion. However, predicting the long-term transformations of communities following an invasion remains a challenging endeavour. We propose an analytical method to predict whether these impacts can be large, as a function of community structure and invader dynamical characteristics, applicable to a wide class of dynamical models. Our approach reveals that short-term invasion success and long-term consequences are two independent axes of variation controlled by different properties of both invader and resident community. Whether a species can invade is controlled by its invasion fitness, which depends on environmental conditions and direct interactions with resident species. But whether this invasion will cause significant transformations, such as extinctions or a regime shift, depends on a specific measure of indirect feedbacks that may involve the entire resident community. Using this metric, we investigate how the complexity, specialization and asymmetry of invader-resident and resident-resident interactions may affect the severity of invasion impacts. Our approach applies to arbitrarily complex communities, from few competing phenotypes to large nonlinear food webs, and hints at new questions to ask as part of any invasion analysis.