Good morning BB
The swash plate is effectively a horizontal thrust bearing that as you say remains stationary but action from the stick causes a gimbal action type of movement so as to transmit the required command from the hand on the stick/pole/Column, I have not taken one apart but it may contain three main parts the stationary base, the one fixed to the rotating blades via the yolk controls and one (or more) intermediate section housing the bearings very simply almost like the thrust bearing on a clutch mechanism .. but very critical to life.