Grazing rights on commons must be safeguarded, for these rights are an historical relic of an otherwise aggressively-advancing culture. Indeed, there are a wide range of benefits from grazing, including the ecological, socio-economic and cultural, though the New Forest – and probably many (or all!) other sites where grazing occurs under tree canopies – is also subject to the damage associated with unrestricted grazing.
Certainly, the number of horses within the New Forest, the unrestricted nature of their movement and the lack of safeguarding measures (and probably food) around veteran trees has resulted in some quite substantial (yet currently rather isolated and sporadic) damage to the beech trees. I would expect much of the damage comes during winter, when the horses are searching for food that is not in such great abundance, and luckily (or not!?) I managed to watch a few horses de-barking a fallen limb and the butt of one particular beech tree (whilst another horse was grazing upon the lower branches of holly), in addition to some recent examples of damage on other beech.