May one brush a dog or other furred pet on Shabbos?
When brushing, a toothed comb — made for the purpose of brushing — is used to extract loose/shedding hairs from underneath the top layer of fur, in order to decrease the amount of shedding an animal will do later. This brings greater comfort to the animal in hot weather, where less fur is less stifling.
Might this procedure be Borer or maybe Gozez?
Brushing hair on Shabbat is tricky because of the melacha of gozez (shearing hair). The original gozez was on animals so brushing animal hair is even closer to the original prohibition! There is no issue of borer here.
Practically, one is allowed to brush human hair only when using a special brush with soft or widely spaced bristles, so that there is a reasonable chance that it will not pull out hair (Peninei Halacha).
The way you describe the activity for animals (shedding loose hair to decrease the amount of shedding later) reads exactly like the prohibition and would thus be prohibited. Since there is likely no difference between brushing Friday afternoon and on Shabbat, I do not see a possibility to permit for tzaar baalei chayim (avoiding animal suffering) either.
See here and here for more details, and remember to consult your rabbi before implementing anything you learn here.
