The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.