I'd be happy to do so . . .
I put a detailed explanation of a how a three stage hand pump works up back on the old Yellow, but sadly it is gone. So I had to dig out my old illustrations. Here is a short explanation of how it works - I figured it all out as I had to trouble shot my Hill Mk2 years ago and made some crude (not to scale) illustrations to show what is happening in the pump as it works.
Remember that these are three stage pumps, so four things have to happen - an intake, and three separate stages of compression - from just two actions of the pump (upstroke and downstroke). Logic says that two should happen on each stroke . . . and that is made possible because these pumps consist of nested tubes for the different stages.
Bottom line, the upstroke delivers the intake and the 2nd stage compression, and the downstroke delivers the first and third stage of compression. The downstroke is much more difficult (on purpose) as it is performing two stages of compression, including the harder third stage. Also, it is the nested nature of the cylinders that result in the long cool down periods required for these pumps as it is hard to get the heat out from way inside them.
I never took apart a four stage handpump, so I don't know how those work in as much detail. While this is for a Hill pump, I'm pretty sure most three stage pumps have the same general construction. Here are the illustrations - the first one shows the layout of the illustration, while the 2nd and 3rd show what happens on the strokes:
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