Intercomponent energy transfer (IET) is a direct consequence of the incompressibility-preserving action of pressure. This action of pressure is inherently non-local, and consequently its modelling must address multi-point physics. However, in second moment closures, pragmatism mandates a single-point closure model for the pressure–strain correlation, that is, the source of IET. In this study, we perform a rapid distortion analysis to demonstrate that for a given mean-flow gradient, IET is strongly dependent on fluctuation modes and critically influences the flow stability, asymptotic states and their bifurcations. The inference is that multi-point physics must be characterized and appropriately incorporated into pressure–strain correlation closures. To this end, we analyse and categorize various multi-point characteristics such as: (i) the fluctuation mode wavevector dynamics; (ii) the spectral space topology of dominant modes; and (iii) the range of IET behaviour and statistically most likely (SML) outcomes. Thence, this characterization is used to examine the validity and limitations of current one-point closures and to propose directions for improving the fidelity of future models.