I made this up a while ago to explain how the pumps work for someone on the Yellow, but never got around to posting it here . . . .
This is based on the Hill pump, but all of the three stage pumps work the same way. I have not had a chance to play with the FX "4 Stage" pump, so I don't know how that one works.
There are three graphics. The first explains the chambers for air in the pump, the second explains what happens on the upstroke, and the third explains the downstroke. I hope it helps explain what is going on in there. It is important to understand is that the some of the O-rings act as one way valves as well as seals by moving slightly to let air pass between the chambers when the stroke direction changes.