The theory of general relativity (GR) displays an array of universality features that are central to the operational use of the theory, such as detecting and extracting information from gravitational waves, while at the same time connecting to foundational issues, e.g., in the gravitational collapse setting or in the high-frequency limit of the theory (in the geometric optics approximation). Quite remarkably, these features can smooth out the details at a certain scale, namely through dissipative/smoothing mechanisms, in a way ensuring some level of predictive power (and simplicity) for the theory even when accurate control of the details is not at reach, either in principle or for operational reasons. We illustrate the role of such universality and dissipative/smoothing mechanisms in a series of examples, with a focus on binary black hole mergers, and we argue for the fruitfulness of (various forms of) asymptotic reasoning for identifying and investigating universal patterns in the GR context. In a broader perspective, we suggest that these latter may provide fundamental insights into the dynamics of (space)time in GR and beyond, in a way that is ‘agnostic’ to the underlying fundamental (e.g., quantum gravity) ontology.