I have black, woven avian netting over my run and I have never had anything other than the occasional chicken get stuck in there....and there was that one time when the goats climbed up there....that was purely due to one end of it coming loose and sagging down where the chickens and goats could climb up it, though.
But, other than that, as long as the netting is held up where the animals *inside* the run can't climb on top it is not a problem.

But, other than that, as long as the netting is held up where the animals *inside* the run can't climb on top it is not a problem.