I haven't read about any homeopathic studies (controlled, double blind) properly done on animals, but I have no reason to believe we'd see any difference between the tested group receiving the homeopathic treatment and the control group receiving placebo.
Most minor illnesses tend to go away by themselves as the body's immune system fights them. If you have a cold and take nothing, it usually passes in 4-5 days. If you take homeopathic medecine, it still passes in 4-5 days, but you attribute it to the medecine.
That's where the common saying "take two aspirins and call me in the morning" comes from. Doctors know that if you give most minor illnesses even as little as 24 hours, you'll feel better.
Longer lasting illnesses also tend to affect you in cycles. You'll have good days and bad days. You'll often take medecine (homeopathic or otherwise) on the days you feel the worst. If you feel better then next day, you'll attribute it to whatever you took.
Most people really want to be validated in their choices. So if they do take homeopathic medecine, they'll look for signs of getting better and tend to excuse any lack of effect. (I didn't sleep well; I was dehydrated; etc.)
Alternative medecines can never lose, because they have an excuse for everything.
If you take AMs and get better, it's because they work.
If you take AMs and they don't work, then it's because you took drugs before and are now "poisoned" or you need to try another AM.
If you take AMs and die, it's because you waited too long before taking them.
Talk to any AM practitionner and try to get them to admit possible failure conditions. It's nearly impossible.
As for infants, I think that the effect is through the mother. Because she thinks she gave some good homeopathic medecine to her baby, the mother is more relaxed and her serene state reflects on the child.
Young kids are also extremely receptive to suggestion. For example, my daughter recently had a cold with a bad sore throat. We had given her some syrup for her throat, but about an hour later, she wanted more. The posology indicated to wait at least 4 hours between doses, and, while I think modern medecine is your best bet when you're really sick, I'm not particularly keen with filling my kids with unnecessary drugs.
So I told her I'd give her a "special medecine." I filled one of her non-spill glass with water and added 3 ice cubes, all the while explaining how the ice would cool the water and that the cold water would soothe her sore throat. I also told her she could have as much as she wanted of this "special medecine." The non-spill glass with ice also made a nice rattling sound when you shook it, adding to the fun.
Later, she told me that the special medecine was working great and that her throat was feeling much better. The next day, her throat still sore, she didn't want the syrup, only the iced water. Her sore throat eventually passed, as those things do, but when she talks about it she says that the special medecine made it go away.
Now, maybe the cold water did numb her throat a little and maybe the water, by hydrating her, lessened the soreness. There probably was some kind of minor effect, but I firmly believe that whatever effect there was, was magnified by her belief that daddy's "special medecine" would help her sore throat.