We used to roll the heavy matting up into a very tight roll, then used a couple of roof rack straps to hold it together. This could then be slung and with a bit of gaffa, to hold the sling in place, it could just be dragged, bit like a torpedo,worked well, or, it did fit nicely into the old Dragon 'darren' tackle sacks, (the thin silver one). Yes its heavy, but it drags well, if you take so care to package it well.
As for it sitting and rotting. I doubt very much it will rot at all, probably be there in tens of thousands of years!! But it doesn't half help when dragging kibbles down a tight passage.