Many farm animals are ruminants (sheep, goats, cows) which can take fibrous feedstuffs (i.e. spent grains) and break them down to carbohydrates. Deer are also ruminants. Related to brewing, their stomachs are essentially large fermenters, with the microbial population responsible for the fiber to carbohydrate breakdown. The stomach itself churns the fermenter (like a stir-bar on a starter) and the animal regurgitates portions of the fermenting feed to chew it/mechanically break it down further. If they aren't actively eating, they are chewing what has already been eaten.
Other farm animals are not ruminants but have digestive systems/organs to break down fibrous feedstuffs (horses). This typically happens in the large intestine, but it not as efficient as a ruminant.
Pigs are monogastric (like humans) so they can't do much more with fibrous feedstuff than we can...it goes in...it comes back out.
Interesting fact about rabbits, since they were mentioned. They also have hind-gut fermentation organs to break down fiber like horses do, but they will also eat their own poop as a way of "sending it through again" for more digestion. They are Caprophagic.