Get Your Ass in the Dungeon

Donkey mini from Otherworld Miniatures (they have 4
variants).  You'll need to put them on a medium base.

Eventually, a Pathfinder player will want to bring his jack or jenny (male or female donkey) into the dungeon. It makes sense. The basis for this desire is the explanation of how these animals behave in the game:

Donkeys and mules are stolid in the face of danger, hardy, surefooted, and capable of carrying heavy loads over vast distances. Unlike a horse, a donkey or mule is willing (though not eager) to enter dungeons and other strange or threatening places. Donkeys and mules have the same statistics as ponies.

The mantra for the donkey is "donkey don't care." Faced with danger, dungeons, strangeness or heavy loads, the donkey does not care.

The key then, is getting your game master on board with the use of livestock in their dungeon. The donkey has been bred as a pack animal for difficult situations for 5,000 years in the real world, so there's no reason it should seem inconceivable in a fantasy setting. Assuming you have honorable intentions, and aren't looking for an extra combat advantage with a cut rate guard dog replacement, the simple question of "Are you gonna mess with my donkey?" should start that conversation.

The Objective Facts

The donkey uses stats borrowed from the pony. That, first and foremost, means a donkey is a medium sized creature. Size is critical, because a large creature in a dungeon is difficult to manage and tends to become a nuisance. A medium based donkey is ideal for a caster, especially a wizard, who tends to stick in the back, avoiding harms way. You can't really manage a donkey remotely (see Handle Animal below), so it's not a great fit for melee characters. Donkeys also don't make the list of familiars or animal companions.

Encumbrance: The donkey has a light carrying capacity of 75 pounds, which allows it to move its full speed of 40. At the next level of encumbrance, 150 pounds, it moves at 30. Bumping that up to 225 pounds doesn't change your donkey's speed, so that's roughly your target (beyond 225 pounds your donkey can't move). You will encumber your donkey to some degree, because there is no food in the dungeon and the donkey must carry it in. Claiming the donkey eats grass and such outside is fine (donkeys have a far more robust gastrointestinal system than horses), but in the dungeon you'll need to hoof in the bags of feed it must consume (more than likely a big 50 pound bag like the one below).

Food is 10 pounds per day, so a week of food is already half his unencumbered weight. My ideal donkey weight is right under 225 pounds, so he's encumbered, but still moves at 30 like most of the average adventuring party. Medium and Heavy encumbrance are both going to knock the donkey speed down to 30 from 40, so load him up to Heavy.

Loading up your donkey with his own provisions, is going to be much of what the donkey carries. Beyond that, you can grab from an enormous list of equipment you would rarely bother with, but might come in handy or add flavor to your character. This is inexpensive and useful at low levels, but loses its charm once you can afford magical bags and the like.

For example, here's what my donkey is carrying:

Air bladder, ale (gallon), animal glue, block and tackle, bread (2), bucket, candle (10), canvas (sq. yd.) (5), chalk, cheese, coffee (per cup) (16), coffee pot, earplugs, feed (per day) (7), firewood (per day), folding pole, grappling hook, grooming kit, honey (per jar), meat (3), mess kit, miner's pick, mug/tankard, oil (7),  pack saddle, piton (2), pot, saw, scrivener's kit, shovel, silk rope (50 ft.), skillet, survival kit, tea (per cup), torch (3), trail rations (14), travel cake mix, twine (50'), waterproof bag.

HeroLab is your friend when it comes to this type of fun/tedium. I've got a bunch of useful dungeoneering supplies along with flavorful items like pots and pans and cake mix. You can certainly live off trail rations, but why not bring some firewood and bake a cake? The total value of this gear is 52gp, and along with 8gp for the donkey, the total replacement cost is a measly 60gp. You could do this at first level with many characters.

Of course, the most hopeful reason to bring a donkey into a dungeon is to carry out vast quantities of loot. Although, the GM is more likely to ding you on encumbrance on your way in, it's nice to have a means to justify your loot carrying capacity on the way out. If nothing else, your donkey can carry the corpses of your fallen companions. As your donkey consume donkey chow, you'll be making room for your inevitable success ... or failure.

Handle Animal: "But what about the Handle Animal skill," you ask? If you're not going to command your donkey to perform a task or push it to do something it doesn't normally do, then you don't need it. Assume the stalwart donkey goes wherever you lead it, on the lead. When you're not leading it, it stands there.

Combat Training: This is not a combat donkey, something to stress to your GM. However, it is safe to fall back on your mantra of "donkey don't care." Combat won't startle or spook a donkey, but neither is the donkey likely to attack unless attacked first. Donkeys were used in WWI by Italian fusiliers to carry their gear, so if modern combat won't spook a donkey, some measly kobolds certainly won't.  As an aside, the donkey was used for food when stores were low, adding one more use for your donkey.

Skills (really Perception): Your donkey has a +5 Perception check. Part of "Donkey don't care" will likely be donkey don't care about Perception checks. If there's something especially disconcerting, your donkey won't communicate it, but might become fidgety or balk when entering an area.

Perception is your only donkey skill of note, and with encumbrance, every other skill on my donkey is zero or worse.

The Subjective Facts

Some basic donkey research reveals a few details:

Animal Relationships. Donkeys and horses get along splendidly, unlike the stories you hear about say, horses and camels. They can interbreed in fact, if you want a little dungeon drama. Donkeys will protect herd animals like sheep by stomping smaller predators, like wolves. Donkeys do not get along with wolves or dogs, and it's safe to say they'll donkey stomp anything small that threatens them. 

Eating, Pooping, Sleeping: Donkeys eat a lot and poop when they eat, so you shouldn't expect donkey scat scattered around the dungeon. Donkeys also sleep lying down, preferably in a clump of hay you've just tossed down for them. Expect to unpack and pack your donkey each day, if such details matter to you.

Stairs. Donkey don't care. Watch videos of tourists riding donkeys down vast stairways in the Grand Canyon. Assume donkeys can manage difficult terrain without trouble.

"Donkey don't care" really sums it up. Sure, a donkey is quickly outperformed by even the lowest level, Type I, Bag of Holding, but that should reassure your GM that you're not trying to get away with anything. An 8gp donkey versus a 2,500 gp magic items certainly seems like a bargain to me.