No.

The Mongol invasions were centuries before the “decline” of China and Korea. I’ll talk about Korea and China since that’s what I know more about.

The Mongol Invasions and the Yuan Dynasty were harsh times for the Chinese and Korean peoples. The Chinese and other ethnic groups were put under the thumb of the Mongol rulers while Goryeo/Korea had to pay a heavy tax in terms of tribute. Mark Peterson argues that along with the Japanese Invasions in the 1590s, the Mongol invasions were so devastating to the Korean people that Korean historical scholarship and archaeology can be divided into pre and post-Mongol invasions

So the Mongol invasions of the Song and Jim dynasties of China and the Goryeo Dynasty of Korea was terrible, but they then formed the Yuan Dynasty of China. The yuan continued the dynastic tradition of China. Confucian scholars continued to operate and an era of relative stability occurred.

This means that after the invasions, China didn’t decline as the Mongol rulers became “Chinese” Rulers to a certain degree. Confucian values were still promoted and confucian scholarship continued to evolve and change in both nations. We also can’t forget that the Mongols linked much of the world with their empires and trade prospered under their rule.

Afterwards, the Mongols were ousted in both China and Korea. This leads to the formation of the Ming and an independent Goryeo Kingdom that would become the Joseon Dynasty.

The Ming would continue to grow economically and develop scientifically, including the development of some Confucian ideas that are similar to liberalism.

In Goryeo and Joseon Korea, we see the introduction of cotton, the ability to create indigenous gunpowder weapons, and (most famously under the reign of King Sejong the Great) the development of Hangul, the Korean alphabet, and rain gauges, and other scientific and engineering works.

The “decline” of Korea is more accurately attributed to the devastating Japanese invasions of the 1590s which killed much of the populationC burned a good chunk of arable land, and created a brain drain as many technicians and artisans were captured and abducted. This was followed up by a famine and the shock of the Manchu Invasions. While Peterson would argue the Manchu invasions were not actually devastating, it can’t be denied that they had a political shock.

In China, Mark Elvin argues for the High Equilibrium Trap. Basically that labor and production, etc just hit a high equilibrium and reduced the demand to innovate. This occurred simultaneously as Taoism and its focus on research and science gave way to a strict Neo-Confucian worldview. This also occurs as the Manchus launch a bloody invasion of China that results in in the west establishment of the Qing. The Qing would “settle” for a more strict Confucian worldview and discourage trade and other policies that would encourage economic growth, like restricting merchant licenses. This last bit is called the Qing Conquest Theory.

In short, the Mongol invasions occurred centuries before the Great Divergence and many of the Great dynasties, like the Joseon and Ming, came afterwards. The decline of the Joseon can be more attributed to the Japanese Invasions and its aftermath combined with the Manchu Invasions The decline of China can be attributed to both the Qing/Manchu invasions and rule along with China just being a victim of its own success due to finding an equilibrium in economics.