In any mixer, diode or FET, the LO, local oscillator power controls the switching elements to on/off. Usually the LO power must be of the order of +10 dBm, typically, to do that.
The RF power (input or output) must be lower than LO, otherwise it would affect the switching function. In upconverters there is a rule that the RF output power , to keep the mixer linear (not saturated), is to be 7 dB lower than LO input. With +10 dBm LO input, the maximum undistorted RF output can be +3 dBm.
Conversion loss is the ratio of RF input to IF output (or reverse), mainly caused by switching resistance of mixer elements (FETs, diodes), and their matching to input and output ports. Good mixers have CL ~ 6 dB typical.