Clear windows onto emergent hadron mass (EHM) and modulations thereof by Higgs boson interactions are provided by observable measures of pion and kaon structure, many of which are accessible via generalised parton distributions (GPDs). Beginning with algebraic GPD Ansätze, constrained entirely by hadron-scale $\pi$ and K valence-parton distribution functions (DFs), in whose forms both EHM and Higgs boson influences are manifest, numerous illustrations are provided. They include the properties of electromagnetic form factors, impact parameter space GPDs, gravitational form factors and associated pressure profiles, and the character and consequences of all-orders evolution. The analyses predict that mass-squared gravitational form factors are stiffer than electromagnetic form factors; reveal that K pressure profiles are tighter than $\pi$ profiles, with both mesons sustaining near-core pressures at magnitudes similar to that expected at the core of neutron stars; deliver parameter-free predictions for $\pi$ and K valence, glue, and sea GPDs at the resolving scale $\pi$ GeV; and predict that at this scale the fraction of meson mass-squared carried by glue and sea combined matches that lodged with the valence degrees-of-freedom, with a similar statement holding for mass-squared radii.