Nanoemulsion-like polymer vesicles (polymersomes) having ionic liquid interiors dispersed in water are attractive for nanoreactor applications. In a previous study, we demonstrated that small molecules could pass through rubbery polybutadiene membranes on a time scale of seconds, which is practical for chemical transformations. It is of interest to determine how sensitive the rate of transport is to temperature, particularly for membranes in the vicinity of the glass transition (T-g). In this work, the molecular exchange rate of 1-butylimidazole through glassy polystyrene (PS) bilayer membranes is investigated via pulsed field gradient nuclear magnetic resonance (PFG-NMR) over the temperature range from 25 to 70 degrees C. The vesicles were prepared by the cosolvent method in the ionic liquid 1-ethyl-3methylimidazolium bis(trifluoromethylsulfonyl) imide ([EMIM] [TFSID, and four different polystyrene-b-poly(ethylene oxide) (PS-PEO) diblock polymers with varying PS molecular weights were examined. The vesicles were transferred from the ionic liquid to water at room temperature to form nanoemulsion solutions of polymer vesicles in water. The exchange rate of 1-butylimidazole added to the aqueous solutions was observed under equilibrium conditions at each temperature. The exchange rate decreased as the membrane thickness increased, and the exchange rate through the glassy membranes was three to four times slower than through the rubbery polybutadiene membranes under the same experimental conditions. These results demonstrate that the permeability through nanosized membranes depends on both the dimension and chemistry of membrane-forming blocks. Furthermore, the exchange rate was investigated as a function of temperature in the vicinity of the T-g of PS-PEO membranes. The exchange rate, however, is not a strong function of the temperature in the vicinity of the membrane T-g, due to a combination of the nanoscopic dimension of the membrane, and some degree of solvent plasticization.