There remains a persistent need for advanced physics-informed and data-driven methods and strategies to address challenges in extracting insights from massive, scale-resolved databases. Physics-informed methods sift primitive variable fluctuations to yield physically more revealing components. Data driven methods are agnostic to the governing equations and generate modes with spatial structure, spectral content and amplitude. Two distinct new strategies are highlighted for multi-scale flow phenomena: (i) application of physics-informed and data-driven methods in sequence and (ii) recovering space-time localization and causal perspectives for transient or intermittent events that may otherwise be lost in data-driven spectral methods. For illustration, we choose a pair of complex propulsion-related problems comprised of impinging jets and multi-stream nozzles, respectively, at ground-test Reynolds numbers. Together, these contain instabilities, feedback, shock interactions and separation, thus forming challenging test beds for advanced methodologies. We show that sequential application of Doak’s physics-informed decomposition into acoustic, hydrodynamic (vortical), and thermal (entropic) components followed by spectral proper orthogonal decomposition, elicits dominant as well as nuanced mechanisms underlying different tonal observations. The second strategy applies conditional space-time proper orthogonal decomposition to highlight transient flow events, distilling causal mechanisms that depend on synchronization of multi-channel information pathways and phase speeds. The results stress the immense scope and utility of advanced post-processing methods.