We review recent lattice results relevant to models of
compositeness for physics beyond the Standard Model, with a focus on work performed by the TÉLOS Collaboration on $Sp(4)$ gauge theories with matter in multiple representations. After motivating the Composite Higgs framework and detailing the theoretical underpinning of the $Sp(4)$ gauge model, we report on the state of the art for the meson and chimera baryon spectrum, including continuum-limit results for both the fundamental and antisymmetric fermionic sectors, as well
as the first application of the HLT spectral-density method to this theory. We then discuss the mixing between isosinglet states, which has no analogue in QCD, and briefly survey complementary lattice studies of symmetric mass generation. We report on non-perturbative calculations of dark pion scattering cross sections that are relevant to strongly interacting massive particle (SIMP) scenarios of dark
matter, and on large-$N$ pion scattering in SU($N$) theories. Finally, we discuss the thermodynamics of the pure $Sp(4)$ Yang--Mills theory and of the $N_f = 2$ model at finite temperature, which are relevant to the production of gravitational waves from early-universe first-order phase transitions.