In regard to the original question, I would put a vague date where the East is ahead (this is obviously generic and differs for individual armies and generals, especially with morale and discipline) overall to around the mid-15th century. East Asian cavalries were much heavier and better disciplined than their western counterparts for most of the Middle Ages. Chinese crossbows are also much more powerful than those used in the west. Even in firearms, the early Ming led the Europeans until about the mid-15th century.
To quote Tonio Andrade (The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History):
During the intense wars of the Yuan-Ming transition, from 1350 to 1450, there were a lot of challenges and a lot of responses, and China’s infantry forces became increasingly focused on firearms, which were used far more frequently and effectively than in Europe at the same time. In the early Ming period, policies prescribed that 10 percent of soldiers should be armed with guns; by the last third of the 1400s, the figure rose to 30 percent, a rate not seen in Europe until the mid-1500s. Historians have labeled the Ming dynasty the world’s first “Gunpowder Empire.”
"By 1380, Ming policies stipulated that gunners should comprise 10 percent of soldiers. Since the total number of soldiers at that period was likely between 1. 3 and 1. 8 million, the number of gun specialists must have been on the order of 130,000 to 180,000, meaning that there were more gunners in early Ming China than knights, soldiers, and pages in France, England, and Burgundy combined. Under Hongwu’s successors, the percentage of gunners climbed higher. By the 1430s and 1440s, it reached 20 percent. By 1466, it had risen to 30 percent. 5 In Europe, on the other hand, it wasn’t until the mid-1500s that gunners made up 30 percent of infantry units"
Andrade Tonio,
The Gunpowder Age, Op. cit. , p 56.
After 1450, Europeans started to utilize arquebuses and swivel cannons, which started to become more and more efficient than the small hand canons of the Ming, although the superiority is far from clear until the 17th century. China also introduced these weapons in the 16th century and were using them alongside native firearms like the three eyed gun and fire arrows. European cannons also fired heavier shots than Chinese ones by the late 15th century, and was superior in siege; however, the Ming and Qing adopted them by the early 17th century and were soon making cannons that were as heavy, and for a brief time, even better made with an iron core. European armies started using flintlocks and bayonets with an emphasis on line infantry formation by the late 17th century, but the Qing used Zamboraks similar to those used in the Middle East for mobile warfare. European star forts were better designed for small scale siege as they only had three sides to assault and can be better defended, but Chinese city walls remained much thicker and bigger than most European city walls. Furthermore, China remained well ahead of European armies in logistics until the 19th century as demonstrated by Kenneth Swope and Peter Perdue. I would really say the Qing fell behind overall only during Napoleonic times, or perhaps slightly before that.