My daughter is 4 years old and occasionally gets constipated. I used to just feed her more fiber, cut out all dairy, and load her up on water, fruits and raisins to try to naturally help her along. Well, that took too long and she suffered for it, which was unacceptable to me. I am not a big fan of using drugs unless absolutely necessary. I asked her pediatrician what I could do and she told me to use Miralax. You can buy it over the counter at any drug store or discount store, like Wal-Mart. It is oderless, colorless and tasteless and it works. It is not specifically for children - it is for anyone, but safe for children. Your child can't overdose on it, so if you don't get results the first try, keep adding to your daugter's drinks until she is able to have a movement. It works like nothing I've ever see - usually works in a couple of hours. My daughter does not even know I've put anything in her drink - it completely disolves and has no taste. It isn't a stool softener, so your daughter may feel some discomfort while going, but she will no longer be constipated.
I also try to limit milk and dairy products in my daughter's diet to about two servings a day and I give her only organic dairy which I'm sure you understand since you raise goats. Good luck. I know how bad you feel for them when they are uncomfortable.