I will present the results of recent lattice studies of the gluon generalised form factors of both hadrons and light nuclei. The generalised transversity gluon distributions are of particular interest since they are purely gluonic; they do not mix with quark distributions at leading twist. In light nuclei they moreover provide a clean signature of non-nucleonic degrees of freedom. The goal of these studies is to inform the development of an electron-ion collider (EIC) designed to access gluon structure quantities including transverse-momentum dependent distributions (TMDs) and gluon generalised parton distributions (GPDs).