First, the Mongols were far from seemingly unstoppable in China. They won many battles, but also lost battles. Their conquest of the Jin took 20 years, and the final destruction of the Jin was carried out in alliance with the Song. Their conquest of the Song took another 45 years.

However, in the west they were viewed as unstoppable until their famous defeat at Ain Jalut. The defeat at Ain Jalut wasn't due to any military innovations, just the non-innovative strategy of attacking a weaker opponent. The bulk of the Mongol army in the Middle East had withdrawn from Syria, leaving just 1 tumen under the command of Kitbuqa. This tumen was supported by allied forces. The nominal strength of a tumen was 10,000, and thus the strength of the Mongol force is commonly estimated as 10-12,000. With the allied forces counted as a tumen (based on the Mongol commander Baidar, a commander of a thousand in Kitbuqa's tumen, being named as an independent commander), the total Mongol strength is sometimes estimated as 20,000. The common estimates of the size of the Mamluk army are similar, 10-20,000.

However, it should be noted that Mongol tumens were often far weaker, and 2-3,000 was quite common. A force consisting of more than one "thousand" (a unit with nominal strength of 1,000, but possibly understrength) was classified as a tumen. Thus, the Mongol force might have been as weak as about 3,000. Mongol losses are variously reported as 1,500 and "most of the army". Contemporaries described the Mongol force as "several thousand cavalry", and noted that the Mamluk force outnumbered the Mongols many times over (and that the common soldiers still expected to lose, due to the yet-to-be-shattered myth of Mongol invincibility). Thus, a lower estimate of Mongol strength appears to be more accurate than a high estimate.Contemporaries reported the Mamluk force as 12,000 strong, and wrote that Qutuz took the "entire strength of Egypt", nominally 24,000. 12,000 is a quite reasonable estimate.

Thus, a balance of forces of about 12,000 vs about 3,000 matches the sources, and the outcome of the battle. Sensible strategy for the Mamluks, and poor strategy for the Mongols, but not innovative.

As for Kulikovo, there are no reliable numbers for the strength of the two armies. Muscovite strength is reported from as "low" as 150,000, up to 1,320,000. Even the low end of these is far too high. Even less is known about the strength of the Golden Horde.

By Ugra River, Muscovy was using firearms in significant numbers (Esper 1969), which might have provided them with an important advantage. But we know too little about the details. Certainly, large numbers of firearms did provide various states with military advantages over steppe forces.

Reference:

Esper, T. (1969). Military Self-Sufficiency and Weapons Technology in Muscovite Russia. Slavic Review, 28(2), 185-208. doi:10.2307/2493223 https://www.jstor.org/stable/2493223