Electronic devices require the growth of high-quality films with low defect densities. High-mobility BaSnO3 thin films have been challenging to grow. Molecular beam epitaxy (MBE) is a low energetic deposition technique and routinely produces the highest mobility semiconductor and functional oxide films. However, stoichiometry control in MBE of perovskite oxides can be challenging. We show that in the MBE of perovskite stannates using a Sn metal source, volatile SnO consumes all active oxygen in the growth environment, leaving behind only Sn droplets at low growth temperatures. No film growth occurs at high temperatures. We discuss a modified MBE approach, which supplies pre-oxidized SnOx. This technique addresses issues in the stoichiometry controls in MBE of ternary stannates related to volatile SnO formation and enables growth of epitaxial, stoichiometric BaSnO3. A second challenge is the lack of lattice-matched substrates. We discuss the effect of lattice mismatch on the growth and transport properties of BaSnO3 by comparing the properties of films grown on SrTiO3 and PrScO3 substrates. We show that the carrier mobility doubles as the lattice mismatch is reduced by a factor of two, indicating that high misfit dislocation densities are the main limiting factor in further improving carrier mobilities. We demonstrate room temperature electron mobilities of 150 cm2 V-1s-1 at carrier densities of 7×1019 cm-3 in BaSnO3 films grown on PrScO3 using the modified MBE technique. We will discuss the prospects for further improving the carrier mobilities and reducing the carrier densities.