Un-gated thermionic cathode RF guns are well known as a robust source of electrons for many accelerator applications. These sources are in principle scalable to high currents without degradation of the transverse emittance due to control grids but they are also known for being limited by back-bombardment. While back-bombardment presents a significant limitation, there is still a lack of general understanding on how emission over the whole RF period will affect the nature of the beams produced from these guns. In order to improve our understanding of how these guns can be used in general we develop analytical models that predict the transmission efficiency as a function of the design parameters, study how bunch compression and emission enhancement caused by Schottky barrier lowering affect the output current profile in the gun, and study the onset of space-charge limited effects and the resultant virtual cathode formation leading to a modulation in the output current distribution.