Higher order topological insulators (HOTIs) are a novel form of insulating quantum matter, which are characterized by having gapped boundaries that are separated by gapless corner or hinge states. Recently, it has been proposed that the essential features of a large class of HOTIs are captured by topological multipolar response theories. In this paper, we show that these multipolar responses can be realized in interacting lattice models, which conserve both charge and dipole. In this paper we study several models in both the strongly interacting and mean field limits. In two dimensions we consider a ring-exchange model which exhibits a quadrupole response, and can be tuned to a C4 symmetric higher order topological phase with half-integer quadrupole moment, as well as half-integer corner charges. We then extend this model to develop an analytic description of adiabatic dipole pumping in an interacting lattice model. The quadrupole moment changes during this pumping process, and if the process is periodic we show the total change in the quadrupole moment is quantized as an integer. We also consider two interacting three-dimensional (3 D ) lattice models with chiral hinge modes. We show that the chiral hinge modes are heralds of a recently proposed "dipolar Chern-Simons" response, which is related to the quadrupole response by dimensional reduction. Interestingly, we find that in the mean field limit both the two-dimensional and 3 D interacting models we consider here are equivalent to known models of noninteracting HOTIs (or boundary obstructed versions). The self-consistent mean field theory treatment provides insight into the connection between free-fermion (mean field) theories having vanishing polarization and interacting models where dipole moments are microscopically conserved.