There aren't only one solution to implement the CMFB, but the common is maybe that they tune one portion of the circuit, others are constant current biased.
In a rail2rail amplifier probably not the best choice to tune a differential pair's tail current, you are feeling the point, because under or above a voltage level one of the tail current sources will be starved, load transistors however are always in saturation.
If you want to implement a tail current controlled CMFB for a rail2rail amplifier that is a challenge, and probably doesn't make sense too much.