The Kerr metric, which describes the geometry surrounding an isolated, rotating compact object, is unique in general relativity (GR), in the sense that all asymptotically flat, stationary, vacuum black holes must be locally isometric to the Kerr spacetime. As such, probing the Kerr metric is one of the best means to test GR. However, because modified theories of gravity are often built such that they contain GR in some limit, the Kerr metric is also common to many of these theories. This means that, in the absence of `smoking-gun' type deviations, a validation of the Kerr metric does not necessarily favour GR amongst all possibilities. But, since perturbations are tied to the gravitational action, disturbances behave differently in different theories. We discuss, using the Newman-Penrose formalism, how the field equation structure manifests in the properties of distorted black holes in non-Einstein theories of gravity. This paves the way for future precision tests of GR in the strong field regime using, for example, gravitational waves.