Moire patterns are periodic superlattice structures that appear when two crystals with a minor lattice mismatch are superimposed. A prominent recent example is that of monolayer graphene placed on a crystal of hexagonal boron nitride. As a result of the moire pattern superlattice created by this stacking, the electronic band structure of graphene is radically altered, acquiring satellite sub-Dirac cones at the superlattice zone boundaries. To probe the dynamical response of the moire graphene, we use infrared (IR) nano-imaging to explore propagation of surface plasmons, collective oscillations of electrons coupled to IR light. We show that interband transitions associated with the superlattice mini-bands in concert with free electrons in the Dirac bands produce two additive contributions to composite IR plasmons in graphene moire superstructures. This novel form of collective modes is likely to be generic to other forms of moire-forming superlattices, including van der Waals heterostructures.