This work investigates the combined effects of a time-dependent effusion rate and a spatially varying viscosity on the thickness profile of a flowing lava. We formulate relatively simple governing equations as a primitive dynamical model for these two influences on lava flow morphology. Free boundary solutions for the profile of the flow and its extent are presented and analyzed for different boundary conditions at the source of the flow and several models for the spatial dependence of the viscosity. The results presented here suggest that time dependence in the flow depth at the source and the form of the viscosity variation are significant influences on the morphology and dimensions of lava flows. Moreover, this analysis implies that time-dependent source conditions may contribute to the scatter in plots of eruption rate versus flow length and disparities between field and laboratory estimates of lava viscosity.