  In the context of massive gravity, bi-gravity and multi-gravity non-minimal
matter couplings via a specific composite effective metric were investigated
recently. Even if these couplings generically reintroduce the Boulware-Deser
ghost, this composite metric is unique in the sense that the ghost reemerges
only beyond the decoupling limit and the matter quantum loop corrections do not
detune the potential interactions. We consider non-minimal {\it derivative}
couplings of the composite metric to matter fields for a specific subclass of
Horndeski scalar-tensor interactions. We first explore these couplings in the
mini-superspace and investigate in which scenario the ghost remains absent. We
further study these non-minimal derivative couplings in the decoupling-limit of
the theory and show that the equation of motion for the helicity-0 mode remains
second order in derivatives. Finally, we discuss preliminary implications for
cosmology.
