  We study the pair production of the long-lived mediator particles from the
decay of the SM Higgs boson and their subsequent decay into standard model
particles. We compute the projected sensitivity, both model-independently and
with a minimal model, of using the muon spectrometer of the CMS detector at the
HL-LHC experiment for ggF, VBF, and Vh production modes of the Higgs boson and
various decay modes of the mediator particle, along with dedicated detectors
for LLP searches like CODEX-b and MATHUSLA. Subsequently, we study the
improvement with the FCC-hh detector at the 100\,TeV collider experiment for
such long-lived mediators, again focusing on the muon spectrometer. We propose
dedicated LLP detector designs for the 100\,TeV collider experiment, DELIGHT
(\textbf{De}tector for \textbf{l}ong-l\textbf{i}ved particles at hi\textbf{gh}
energy of 100\,\textbf{T}eV), and study their sensitivities.
