Why is that? Does one of my stats have to be good (int or skill)?

Nope. All the monster SUCK at following orders.

Glad to know it's not just me.

Curiously enough, I haven't had much trouble directing my monsters on errantry this time round, except for when they're exhausted or if they've been beaten by a stray; I found they respond quite well when you have them turn 90 or 180 degrees, so long as you repeatedly tap in the general direction you want them to go.

Didn't loyalty or wits have some sort of effect on whether or not they'll follow your instructions?

I can honestly say that if loyalty and wits played a factor, it didn't help much. Monsters are definitely more obedient when those factors are high, but still are subject to the horrbile errantry engine. Tiredness definitely plays a role. The longer they go, the less responsive they become.
My real frustration is in the collision detection. When they get remotely near a wall, they abandon their orders and walk the opposite direction. Getting them to walk straight is just about impossible. Add all the distractions from crystals and it gets pretty annoying. The volcano Errantry has to be the most frustrating of all.

Well, I'm not sure if it's just me or not, but from my experience it seems like tiger breeds seem to follow commands quite a bit more often than other breeds....

Granted, I haven't explored this theory enough to be sure, but both my tiger/abyss and tiger/zan followed commands roughly 90% of the time, wheres my zan/zan and xenon/xenon only listened to orders 40-60% of the time at best.

All four monsters had the highest level of wits and loyalty, and all were above rank A while on Errantry.

Seriously though, is this just me? Cuz I swear I had to bring one of my tigers out of retirement to finish the volcano map and, amazingly, it cleared it with at least a third of it's life left.

Fatigue also affects how well they listen. The best way to keep the monsters fatigue level down is to avoid monster fights.

Breed definitely plays a role, for sure. Some breeds just act like real putzy boys. But at the same time, fatigue will make it worse.

I could be mistaken, but I'm pretty sure your monster's intelligence stat plays a huge role in how well they listen. My Tigers (most of which have crappy intelligence, like 150 and below) are HORRIBLE at listening during errantry, but my other monsters (Phoenixes, Ogyus and Pooch) listen very well, and they have 300-500 intelligence.

Nope. It doesn't.

My bad monsters are just as bad when they have 200 int as when they have 999 int.

My theory is it depends on wits, loyalty, Int and Grade. I've Raised 2 Piroro Grade B Max Loyalty and Wits. 420 Int. and it would barely listen. I raised a Bree ( Piroro/Falco ) Max wits, Max loyalty, Grade A and 534 Int. and it listened perfectly until I hit around 1/3 of Life left. After that it seemed the closer my Life got to 0 the less it listened to me. This was just my recent experience.