Two major forces shaping evolution are drift and selection. The standard models of neutral drift-the Wright-Fisher (WF) and Moran processes-can be extended to include selection. However, these standard models are not always applicable in practice, and-even without selection-many other drift models make very different predictions. For example, "generalised Wright-Fisher" models (so-called because their first two conditional moments agree with those of the WF process) can yield wildly different absorption times from WF. Additionally, evolutionary stability in finite populations depends only on fixation probabilities, which can be evaluated under less restrictive assumptions than those required to estimate fixation times or more complex population-genetic quantities. We therefore distill the notion of a selection process into a broad class of finite-population, mutationless models of drift and selection (including the WF and Moran processes). We characterize when selection favours fixation of one strategy over another, for any selection process, which allows us to derive finite-population conditions for evolutionary stability independent of the selection process. In applications, the precise details of the selection process are seldom known, yet by exploiting these new theoretical results it is now possible to make rigorously justifiable inferences about fixation of traits.