  This paper reports a study of the experimental signatures of hierarchical
sectors beyond the Standard Model characterized by a flavor-violating heavy
vector and a set of light pseudo-Goldstone bosons $a_{1,2}$, spanning a large
range of lifetimes. The non-minimal scalar spectrum triggers novel $B$ decays
into multiple leptons that would have escaped the reach of current searches.
Novel displaced vertex analyses at LHCb are therefore discussed to probe the
hierarchical new physics, extending the scope of the tracking system of the
detector. Additionally, the reach of the proposed CODEX-b experiment is studied
as well. By exploiting the use of tracks only reconstructed in subsystems of
the detector at the high-level LHCb trigger, ${\mathcal{B}(B_s^0\to a_1 a_2) <
10^{-8}}$ and ${\mathcal{B}(B^+\to K^+ a_1 a_2) < 10^{-9}}$ could be reached in
the muon channel across seven orders of magnitude in the lifetime of the
Goldstone bosons. Correspondingly, heavy-light particle couplings of order
$\lesssim 1$ could be tested, potentially ruling out composite Higgs scenarios
where the heavy and light sectors couple strongly.
