For example, would a Chinese army fighting off Mongol hordes of horse archers with gunpowder weapons find themselves at a disadvantage because the weapons took so long to load? Or would these gunpowder units have been small, specialized units that fought alongside their conventional army?
The development and introduction of gunpowder weapons is a rather complicated history. There's not a sudden changeover from bows to guns. Also, other technological and organisational changes make comparisons difficult.
In European armies, handguns and bows coexisted in armies for a long time. The characteristics of early guns and crossbows were fairly similar. Sometimes units would even be mixed between the two. It took centuries for guns to gain a decisive advantage. Even then, England still used the longbow into the Tudor period.
So I believe it's rare in the late medieval period or early renaissance to see a 'gunpowder' fighting a 'conventional' army. More likely you'll see an army with more guns and fewer bows fighting one with more bows and fewer guns.
Moving into the renaissance, you see the development of 'pike and shot' tactics. These had around blocks of pikemen (and sometimes swordsmen or halberdiers) supported by units of musketeers, plus cavalry (who, early on, might still be heavily armoured). It's only with the development of the socket bayonet that pikemen began to fall out of use and we see armies where all the infantry have guns. (And I have heard there was some talk of raising pike units in the Napoleonic wars, though I don't know how true this is.)
It's true that European armies did not gain a crushing advantage over non-European ones until the introduction of breech-loading rifles, and then repeating rifles, in the latter half of the 19th century. (Note that most non-European armies they faced would have at least some gunpowder weapons, from well-organised Sikh armies to harassing fire from Afghans with rifles that outranged British muskets.) However other factors gave them powerful advantages. Many British colonial battles in the first half of the 19th century seem to be won by superior discipline rather than superior firepower. Again, we have to account for disparities in training, numbers, etc. when looking at battles. (So while Isandlwana was a defeat for the British, they were also outnumbered perhaps 20 to 1. It probably wasn't their rifles that let them down.)
In the specific case of China, I have seen it argued that the fact the main threat came from light mounted forces was one reason that firearms did not become more advanced and more common. A bow could fire faster (or a crossbow just as fast) as an arquebus or musket, could be more accurate, and was perfectly adequate at killing or injuring someone (or their horse). It was in Europe where the widespread use of heavy armour, and also the prevalence of siege warfare, made firearms attractive.
All that said, I'd be very surprised if there weren't instances where guns were at a clear disadvantage - fighting in wet conditions is an obvious case where gunpowder weapons, and particularly matchlock guns, had big drawbacks.
(Main source here is Weapons and Warfare in Renaissance Europe by Bert S Hall.)
Actually yes. This happened during the Battle of Plassey. Nawab of Bengal Siraj-ud-Daulah was in a conflict with Lord Clive of East India Company over taxation overreach by the Company. The fight was raging on for days and Nawab was clearly in a position of advantage. And then, the rains came. The Nawab's army didn't anticipate this and didn't have a tarpaulin to cover the gunpowder. The rains went away and so did Nawab's advantage, as the British did manage to protect their weapons. The tables turned, the Nawab lost and India's colonization formally began.