Theory in ecology and evolution often relies 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 a general analytical method that uses both resident community and invader dynamical features to predict whether an invasion causes large long-term impacts on the invaded community. This approach reveals a direction in which classic invasion analysis, based on initial invasion growth rate, can be extended. Indeed, we explain how the density dependence of invasion growth, if properly defined, synthetically encodes the long-term biotic transformations caused by an invasion, and therefore predicts its ultimate outcome. This approach further clarifies how the density dependence of the invasion growth rate is as much a property of the invading population as it is one of the invaded community. Our theory applies to any stable community model, and directs us towards new questions that may enrich the toolset of invasion analysis, and suggests that indirect interactions and dynamical stability are key determinants of invasion outcomes.