The two-dimensional, regular, hexagonal foam is a common benchmark system for studies of foam rheology under imposed shear. Traditionally, the hexagonal foam has been studied under quasistatic shear deformations. Here, however, nonequilibrium systems are considered. Specifically, the hexagonal foam is assumed to depart from physicochemical equilibrium (surfactant coverage varies and hence surface tension varies between films), but to remain in static mechanical equilibrium (as a consequence, films in the hexagonal foam remain straight, simplifying the geometrical description considerably). Regardless of the mechanism for departing from equilibrium, topological transformations (during which certain films in the hexagonal structure shrink to zero, and bubbles exchange neighbors) tend to be postponed compared to an equilibrium foam. Even when the rate of imposed shear is small, significant departures from physicochemical equilibrium are still observed on the approach to and in the immediate aftermath of a topological transformation. The nature of the relaxation post-topological transformation depends on the variation of film tension with surfactant coverage. It may be almost entirely mechanical (in the case of weakly varying film tension) or almost entirely physicochemical (for a strongly varying film tension). As the imposed shear rate increases, models incorporating weak departures from physicochemical equilibrium prove inadequate to predict the criteria for topological transformation: far-from-equilibrium physicochemical properties must be considered also. (C) 2012 The Society of Rheology. [http://dx.doi.org/10.1122/1.3687442]