We investigate the potential reach of a search for a long-lived/prompt dark vector boson $Z_D$, also called dark $Z$, and a prompt dark Higgs boson $h_D$ through exotic decays of the observed Higgs boson $h$ into either $Z_DZ_D$, $h_Dh_D$, or $ZZ_D$ with $Z$ being the hypercharge gauge boson. The $Z_D$ production through the Higgs portal is completed via one of two mechanisms, kinetic mixing of $Z_D$ with $Z$ and the mixing of $h_D$ with $h$. All production modes of $h$ are considered, while the branching fractions are calculated in Monte Carlo simulation using the {\textsc{MadGraph5}}\_aMC@NLO v2.7.2 framework. We focus on a final state of multiple dimuons, displaced up to \mbox{7500 mm}, where the muons can be reconstructed without vertex constraint using data from ATLAS and CMS detectors to be collected in Run~3. Integrated luminosities of 137, 300, and 3000 fb$^{-1}$ for Run~2, Run~3, and high luminosity run (HL-LHC), respectively, are used for estimating the expected search sensitivity of the Large Hadron Collider to each of the decay modes.