If trained in rope and respectful of halters, they can be trained in either nylon web or leather aswell.
How do I know? Well let's just say my horse used to destroy nylon and leather so I stuck him in a rope halter for training and now he respects both of them :)
I would NEVER EVER trailer with a rope halter though. Either nylon web or leather (my preference.)
If you turn out, never do so in a rope halter because they don't break. And if they don't break, then the horse is going to be stuck in whatever situation they got themselves into.