How can you work out how much distance you travel towards a position directly below you as you move around a circle?
You need to define a suitable reference system for your question, such as a three-dimensional earth-fixed cartesian system with its origin at your point of departure, and its Z axis aligned with the local vertical at same. Then your "distance [...] towards a position directly below you" is simply the value of Z at your destination, and the "distance you travel [along Z] towards a position directly below you" can be found by integrating |Z| along your route.
Does this answer your question? Sorry, but the phrasing in your original post was a bit vague so I'm just having a best guess at what you were trying to ask.