Old thread, but I surfed in on it... I'm debating on throwing a leash on the dog at the moment and letting him indulge his curiosity from inside the goat pen.
Lots of good thoughts, I will add a tad;
There's a bit more to it than training. :-) In a perfect world we could completely defeat instinct through repetition, but in all reality there are more factors than how badly you want it to work in play. Our dog's a Mcnab-bully cross, so his Mcnab side instinct is to run, get close and drive them, but to stay out from under their feet. His obsessive love/fear of them is what would keep him alive herding cows. Experience might turn that fear to caution or even skill were it not for his bully side, but meantime he's just neurotic. There is no degree of training that will take away that drive. Add the terrier and double that. Short of electricity and I find that to be solving a non problem, since the goats just find him stupid, since he runs towards a fence and skids to a halt about fifty times a day. They graze the fence line just to mess with him now. He's having kittens and they're playing goaty games on the other side of the fence to freak him out more - there is no undoing what he is. Goats don't always "flock" to way other stock animals do. They've got a stubborn streak and flight isn't always the choice. Might be less an issue though if your dogs don't have working breeds in them. I'm thinking a less high drive dog would probably be better able to be a companion to them than my crazy.
How tolerant are the GOATS. I worry more about the DOGS than the goats. We have a small herd of nubian/ nigerian dwarf crosses - roughly same shoulder height as the dogs, and I will have to say, the buck will give any dog second thoughts about whether they want to make the goats run. Our buck will let himself out and head for the back yard if I don't get them in the pen they prefer during the day, and if I'm slow about it he'll take himself to visit the dogs. He plays with his lady friends often and sometimes a bit rough. He'll take a running start and give em a good slam if there's high value treats or scritchies to be had.
The girls are goats. They're built to take what he dishes out, but the dog is not. If he did to the dog what he does to the girls, we'd be looking at vet bills and at the very least broken ribs.
The dog has self preservation on his side, in their one accidental meeting - free range goating incident, I wasn't up and at em as fast as the goats would have liked; and our boy ended up in the back yard with the dogs. When I got to him, the dogs were on the other side of the yard. They wouldn't let him past a certain point, but they weren't getting within twenty feet of him, so I have a sneaking suspicion he may have been teaching them about goats. The dogs wouldn't get in range of horns or feet. Now that he's on his side of the fence he's been doing that horizontal goaty jump run on the fences threatening the dogs, so I think he thinks he has street cred now. Everybodys a tough guy. haha.
We had a boer wether and he decided to visit the dogs - he could pop any latch, and with size on his side, when he came a calling, it was an unquestionable run to rescue the dog.
The dog is notoriously stupid though. He got skunked repeatedly because he couldn't get it through his head that the skunk wasn't intimidated by him and wasn't going to run and kept repeating the error. if it moves, he's going to run at it. Once he starts running, the only things in his head that still work are instinct. Run, bark, don't get stepped on.
The nubian cross girls might tolerate him trying to herd more, but they prefer to follow than to be chased when we move them. If your goats know they're goats - as opposed to horn people, and have an established group and pecking order, there's a pretty good chance that the head goat may decide to get cranky about being pushed around by this obnoxious "goat." Seasonally exacerbated if you have intact male goats.
They goats get along hunky dunky with the ducks though. I'll put them in together just to see the puzzled looks on the goats. The ducks are so bossy the goats will follow them.