Interaction between localized plasmons in isolated proximal nanostructures is a well-studied phenomenon. Here we explore plasmon-plasmon interactions in connected extended systems. Such systems can now be easily produced experimentally using graphene. However, the mechanisms of plasmonic interactions in extended systems are not well understood. We employ finite-element methods to study these interactions in graphene superlattice nanoribbon arrays with a periodically modulated electrochemical potential or number of layers. We find a rich variation in the resulting plasmonic resonances depending on the dimensions, the electrochemical potentials (doping), and the separation of the nanoribbon segments, and we demonstrate the involvement of both transverse and longitudinal plasmon-plasmon interactions. For example, unlike predictions based on the well-known "orbital hybridization model," the energies of the resulting hybrid plasmonic resonances in the extended system can lie between the energies of the plasmons in the individual components. Our results demonstrate that the plasmonic spectra of graphene superlattice structures can be easily adjusted, continuously tuned, and used to enhance optical fields in the infrared and terahertz regions of the electromagnetic spectrum.