I have a mohair one. I use it for my side saddle since it gives, but not as much as elastic ends.
Since my horse is girthy, I thought I'd try it with my astride saddle, you know because it's supposed to be so comfy and all. Well, it slips and lets my saddle ride forward on the withers, which causes my horse to buck.... so nope, won't be trying that again. In fact, I'm not to keen to use it with my side saddle again either.
I discovered nylon string girths eons ago while in Pony Club. Those inspectors at rally must have thought our team sure had a buch of gall proned horses as my whole team converted to them. We actually used them because they were cheap enough to use only as "show tack" and since we just threw them in the washing machine it was one less peice of leather tack to obsess over getting clean.
I like them and still keep a few in different sizes around. Being wider I think they do a better job displacing tension--unlike a shaped girth which can only carry tension as wide as its narrowest point. (This would be the reason I used a tri-fold for many years!) Since it does not have elastic you do need to be careful that you get it tight enough. A few extra minutes and efforts of tightening it before you get on is important.
Pros: super easy to wash, cheap enough to have several. Almost impossible to over-tighten, consequently, a little easier on your billets. Softer and flexible and conforms, somewhat, to the shape of the horse. Can be a good solution for strangely shaped horses, especially strangely shaped horses that get girth sores because of their strange shape.
Cons: you almost need to buy them long enough so that the buckles reach the edge of your saddle pad (if it's a short dressage-type girth) because since there's nothing behind the buckles, they can sometimes pinch. Mohair stretches - you'll wash it and it will appear to shrink, you'll tug to get it attached, then 10 minutes into your ride it will be loose.
Hate them, no give at all. Many years ago had a trainer who was very pro string girths so I used one for a while, my poor horse suffered through and at that time I was too ignorant to realize that my horse wasn't moving as well as she could. When I did change to a leather girth with some elastic, she started breathing easier and moved better. Never again.
I used one on my round QH because it did not slip back. Worked well, easy to wash, but I did catch a spur in it for a couple of seconds once while jumping. Nothing happened but it was a disconcerting moment.