I was watching a video progression on how to teach the butterfly pull up. One thing that stood out was the “swooping” motion of the lower body to drive the upper body up.

I’m  a visual learner, and the closest thing I could imagine that illustrates the mechanics behind this movement is the transferring of linear force into rotary force – like in a 4 stroke engine. Now in the pull up it works the opposite direction, but the video still does a great job of showing this.
https://www.youtube.com/watch?v=OGj8OneMjek

Focus on the the crank shaft as the lower body, this is where the “swoop” or rotary movement occurs. If you imagine manually moving the crankshaft, you’ll see how this turns into linear force which will drive the piston/rod assembly straight up and down.

While you end up having a little more movement in your torso from front to back, the general principle still applies: Use rotary force from your lower body and transfer it into a linear force to drive your upper body up and down while controlling the forward/aft motion to maintain solid tension and torque throughout the muscular kinetic chains.