I don't have any livestock, but neighbors on both sides of me have had horses at one time or another. But the neighbors are complete opposite situations in terms of how they care for the animals.
The property on one side was owned by an "absentee landlord". There was no house on the property and he just rented the place (8 acres) out to people to keep their horses on. Problem is, there was no grass and the property is mostly wooded. To make it worse, the renters kept three horses there and only visited a couple of times per week. Since they didn't own the place, they paid no attention to the fence or what the horses were doing to the property (or my property).
The end result was that they chewed what little grass there was down to the dirt, which caused erosion to the point that their property is now two feet lower than mine, also causing the fence (which is mine) to start collapsing, so I now had to try to fix the situation. This is a 5 wire fence, so the horses were also reaching over/through the fence and chewing my grass down, causing the same erosion on my side. The horses also tore the fence down and I was constantly finding them on my property because it was the only place they could find food. I wound up having to install wire mesh fencing over the 5 wire to keep them out, which of course creates other issues, particularly regarding keeping brush cleared from the fence line. Basically, huge maintenance problem for me, and nobody over there gave a damn.
On the other side, I had a single lady, owned 6 horses, same type of property (trees, no grass) but since she lived there and was responsible, I had very little trouble with the animals because she actually cared for the them, made sure they were properly fed and knew where they were most of the time. Well, almost. One time she let them run free for a couple of days back in the woods, and caused the same erosion problem on that side. But once she figured out what was happening, at least she took steps to stop it. After that, she kept the horses in a corral up by the house and fed them hay.
The point of all this is, I felt completely different about the situation when it was obvious to me that the owner of the livestock gave a damn about the impact to the neighbors versus the one who didn't.
Now, neither of the neighbors have livestock and life is good.
As for the original post, wow, I can't imagine anyone acting like that to someone they've never even met, especially a new neighbor. About all you can do is ignore them, I guess.