It makes sense to me that the Chinese, the inventors of gunpowder weaponry, who also spent a lot of their history fighting off mounted invaders like the Mongols, would have used this technique when defending themselves from their enemies' more advanced cavalry. I am by no means an expert on Eastern history, and I just thought that this would make a lot of sense.
The pike and shot tactics we think of (like the Tercio, the Dutch battalion, or Gustavus Adolphus's linear tactics) evolved in Europe, for European battlefield conditions, so there isn't going to be a 1:1 equivalency between European and Chinese tactics, but they're not so different as to be incomparable.
If by pike and shot, one just means the use of spears and (usually) matchlocks in concert, there is some evidence that late Imperial China was using these kinds of weapons together in an organized fashion. Pictorial evidence, like in the paintings of the Qianlong Emperor reviewing the Eight Banners, shows two ranks of soldiers armed with spears twice their height, standing between a light barricade and a platoon of musketeers, formed up five ranks deep.
https://upload.wikimedia.org/wikipedia/commons/9/9b/Emperor_qianlong_blue_banner.jpg
Also, while people tend to stereotype the Mongols and other nomadic cavalry centric armies as being comprised entirely of horse arches, this isn't always the case. During the time of Genghis Khan, for example, the Mongol contingent of the army was usually split 6-4 between light mounted archers and heavier lancers, and during the early conquest period, the Manchu often directed head on charges towards the Ming lines, which were repulsed with heavy casualties at the Battle of Dalinghe.
Furthermore, we do have textual evidence for the coordinated use of firearms and cold steel; volley fire (alternatively known as countermarching, or fire-by-rank) is usually thought of as a Dutch (or perhaps Japanese, though the evidence is disputed there) innovation, when there's evidence of its use in China back to the Tang Dynasty. Manuals of the period would show the disposition of a crossbow armed platoon, dividing the men up into stringers, loaders, and shooters, and works from the mid-late Ming apply similar techniques to firearms, and furthermore describe their use with traditional weapons.
https://40.media.tumblr.com/a34a9805ee1093c7d48eb7da9a734c2c/tumblr_inline_nwdompsbu81rf4l09_540.png
Here, you see the Ming writing about how they organized firing in volleys, listening for specific signals from their commanders, waiting until the enemy was within optimal range, firing only when ordered and in volleys, and knowing when to retire and turn the fighting over to those armed with cold steel.
Now this alone isn't ironclad proof of Chinese tactics during the period, since these aren't detailed descriptions of actual battles, -countermarching was described in Europe for years before Maurice of Nassau put it into practice at the Battle of the Dunes- but they do show that the Chinese were aware of the potential uses for their gunpowder weapons.
Pike and shot was a refinement of the Swiss pike wall in response to threat from the French heavy cavalry formation. The Swiss like in particular had so much discipline and cohesion that they could withstand a straight up heavy cavalry charge and also move on the offense. As gunpowder became more effective, a combination of pike and shot allowed for a very flexible deployment that can both take the offense and the defense.
Mongol cavalry fought very differently from French-style heavy cavalry. The Mongols fought as mounted archers, whereas the French heavy calvary did not. So it would not make sense to use a pike block to counter mounted archers. The mounted archers could just shoot at them while out-maneuvering a pike block.
Pike and shot was only one method of countering a mounted opponent. There was a second method developed in Europe that proved just as effective at stopping cavalry - the wagenburg, a ring of wagons festooned with firearms equipped soldiers, pikemen and cannon. Such tactics are normally associated with the Hussites of Bohemia but the idea spread eastward with the Ottomans picking it up after the Battle of Varna and then transmitting that knowledge to the Mughals who used it to conquer Northern India. Additionally, these methods were used by the Russians, the Poles and the Cossacks as well.
The Ming also utilized similar tactics, though they were most likely independently developed. This innovation may be traced back to the general Qi Jiguang. Qi Jiguang was famous for crushing Japanese pirates that had plagued China's coasts. One innovation developed during this campaign against the pirates was the Mandarin Duck formation which featured a combined force of two sword and shield bearers, four lancers, two soldiers carrying tridents which discharged gunpowder propelled arrows and two men carrying bamboo trees to confuse the enemy. After he was transferred to fight the Mongols, he developed a so-called battle wagon. These battle wagons would form a solid defensive walls against any mounted opposition.
The wagon was equipped with light cannons and crewed by 20 men. Half the crew would remain with the wagons while the rest would be equipped like the soldiers of the Mandarin Duck formation with lances, shields, swords and firearms. When the Mongols had broken themselves on the wagons, the infantry would emerge to mop up, though they were instructed to remain close to their wagons. Heavier cannons were also carried on four wheel carts to supplement the firepower of the battle wagons. Qi Jiguang envisioned a brigade of 3000 cavalry, 4000 infantry, 128 heavy wagons and 216 light battle wagons though ultimately these plans never came to fruition. Page 74-75 of Ray Huang's 1587, a Year of No Significance shows Qi Jiguang's battle wagon formation as well as the Mandarin Duck formation.
