  The hierarchy of the coupling strengths in a physical system often engenders
an effective model at low energies where the decoupled high-energy modes are
integrated out. Here, using neutron scattering, we show that the spin
excitations in the breathing pyrochlore lattice compound CuInCr$_4$S$_8$ are
hierarchical and can be approximated by an effective model of correlated
tetrahedra at low energies. At higher energies, intra-tetrahedron excitations
together with strong magnon-phonon couplings are observed, which suggests the
possible role of the lattice degree of freedom in stabilizing the spin
tetrahedra. Our work illustrates the spin dynamics in CuInCr$_4$S$_8$ and
demonstrates a general effective-cluster approach to understand the dynamics on
the breathing-type lattices.
