Well, unfortunately there are 2 diffierent uses of the term 'pull through' which can understandably lead to confusion if unfamiliar! In the case of the discussions on the best knot / dodgy pull throughs in the Peak District, what we are actually talking about is reaching a belay point / high level passage from the ground which can't be rigged on the way down.
In this case, because you have to pull a knotted rope through the belay, it tends to get stuck- whereas on a 'pull down', it can be rigged without the need for a knot....
So.... There is a large ring bolt or suitable belay at the top of the pitch, but you are at the bottom, needing to get a rope up there and secure. Through the rope at the top is a thin polypropylene cord, which is tied in a loop, coming down from the single upper belay to a pair of bolts a way apart at the bottom, forming an 'A' shape. One end of the caving rope is tied to an end of the poly rope, and the opposite end of the polypro is pulled upon- by magic the caving rope goes up, through the belay and comes back to the bottom where it is tied off to the bolts. Hey presto! the pitch is rigged, prusiked up and abseiled back down, and the caving rope can be pulled the opposite direction and retrieved. You have to make sure you only use the one correct side of the rope!
Does that make sense? :?