if they are not feeding their chickens then that is wrong - so are you sure about this? Have a word with them and explain what is happpening and perhaps let them know that you were thinking about getting a dog (whether you are or not ) so for their hens safety it might be best to clip their wings or get your neighbour to put up higher fencing.
I do have a dog - a great soppy mutt who doesn't give a hoot about the chickens and will walk straight past them without even looking. Good in one respect, but not in another. Strangers, foxes and rabbits are another matter entirely! Another dog might be an idea. A friend has a litter of terriers (or terrorists as he calls them) so a good puppy would not be a problem to find.
Yes, I am sure that they do not feed the birds. There is a feeding trough in their pen but in the 18 or so months I have been here, there has never been any food in it I pass it several times a day and would have seen, I am sure. The chickens forage where they can. A large manure heap by some stables has been a favourite source of food for them as it has been full of worms etc, but that has now been spread on a nearby field.
As mentioned the neighbours have been unhelpful when I have approached them, but perhaps a gentle hint about the damage and the cost of it will help to concentrate their minds. Luckily I have a solicitor in the family, so if it comes to hints which are less than gentle, that may be useful. I cannot blame the chickens, they are hungry. Another solution may be to take them over myself. I have space for runs and housing and I could then quite legally clip wings. One thing, though, They do not, as far as I know, lay eggs. Probably they are too underfed to do so. It would not stop the people obtaining yet more chickens, though, so I might solve one problem to find a similar one.