Why not have a word with your local rights of way officer? Whilst you will be aware that you have a legal duty to maintain gates and stiles etc on rights of way, most councils will either undertake it for you or heavily subsidise the work, especially if a recognised walk passes over your property. If they are willing you can ask for a self-closing gate to be installed. The mechanism is set slightly at an angle and, when opened, it drops back down into the shut position. It can be used on bridle gates, kissing gates and fieldgates with equal success.

There is no excuse for any walker destroying any of your property, stealing stuff or leaving litter (or any other type of undesirable behaviour at all). I can't condone this at all. But, if this results from having a promoted walk through your property then the council must suggest remedies. Contact them and let us know the result.