Motivated by the success of the flavour physics programme carried out over the last decade at the Large Hadron Collider (LHC), we characterize in detail the physics potential of its High-Luminosity and High-Energy upgrades in this domain of physics. We document the extraordinary breadth of the HL/HELHC programme enabled by a putative Upgrade II of the dedicated flavour physics experiment LHCb and the evolution of the established flavour physics role of the ATLAS and CMS general purpose experiments. We connect the dedicated flavour physics programme to studies of the top quark, Higgs boson, and direct high-p_T searches for new particles and force carriers. We discuss the complementarity of their discovery potential for physics beyond the Standard Model, affirming the necessity to fully exploit the LHC’s flavour physics potential throughout its upgrade eras.