The regulation of cell division, cell sizes and cell arrangements is central to tissue morphogenesis. To study these processes, we develop a mechanistic hybrid continuum-discrete mathematical model of cell dynamics that has advantages over previous approaches. This model borrows ideas from statistical physics, materials science and applied mathematics and follows the framework of dynamic density functional theory. This approach provides a strategy for coarse-graining systems of stochastically interacting particles. By appropriately accounting for cell size and shape variability, we obtain a system of continuum equations that are able to capture plastic, viscoelastic and elastic deformations in the clusters while providing single-cell resolution. The discrete component of the model implements cell division and thus influences cell sizes and shapes that couple to the continuum equations. We present efficient numerical methods for solving the coupled systems of equations and validate the model using recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with the experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with the emergence of short-range ordering and a constant-thickness proliferating rim of cells at the cluster edge. We then discuss extensions of the model to account for different cell types, including stem, progenitor and terminally differentiated cells, that are needed to model stratified epithelial tissues. Finally, we describe how this framework can provide a rational approach for upscaling from the subcell and cell scales to tissue scales to obtain new, tissue-scale governing equations.