  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/HE-LHC
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.
