High-gradient radio-frequency electron guns (RF guns) produce significant field emission (FE) current (also known as dark current) during operation, which can impact the normal operation of the accelerator. Since the primary sources of FE current are challenging to predict and observe directly, a detailed investigation of FE current characteristics is necessary to develop strategies for mitigating its impact. In this study, we systematically study the transport and distribution characteristics of FE current from two typical emission regions: the cathode and the cavity rim of the RF gun. The simulation results show that FE currents from these regions have similar energy distributions. However, high-energy electrons from the cavity rim are lost due to collisions with the cavity wall, while electrons from the cathode are more effectively transmitted downstream. Given that the FE current has a large transverse momentum, which makes it prone to loss to the vacuum tube wall, controlling the solenoid field strength is crucial for acquiring more detailed information about the FE current and optimizing its management.