This is an interesting question because it highlights what I feel is a common misconception about exactly when European monarchies "fell". I will focus on your specification of a movement which "demolished a monarchy". While there are many examples to be used, I will be focusing on France and the United Kingdom, which I know best.

Many people like to focus on the French Revolution as the "start" of Democracy in France, but the reality is much more complicated. The revolution happened in the final decade of the 18th century, but that was not the end of the monarchy in France. Napoleon claimed the title of Emperor of France from 18-04-1815. He was followed by a restoration of the Bourbon Dynasty (1815-1830), then the July Monarchy (1830-1848), and then the Second Empire under Napoleon III (1852-1870)

Pro-Democracy sentiment had developed in China had developed by the late 19th century. In 1905, a group of Imperial officials traveled to Japan, Europe, and the United States. According to Charles Holcombe, "After their return, a nine year program of constitutional reform was announced in 1908 that included the installation of elected provincial representative assemblies by 1909 and a promise of gradual transition to 'self-government'". The people of China were not satisfied with this promise, and overthrew the Qing Dynasty in 1912. A Democratic government was established in China, but failed in 1916 and was succeeded by a series of "warlord states", which lasted until 1928. The country was reunified by Chiang Kai-shek in 1928, just in time for World War Two, and the subsequent Communist Revolution.

The United Kingdom spent the majority of the 19th century under the reign of Queen Victoria, who died in 1901. During this time the United Kingdom itself was technically ruled by a democratically elected parliament, and had been for quite some time. However, Queen Victoria still held an incredible amount of social influence, and while her political power was no where near that of an absolute monarch, it wasn't negligible either. It should also be noted that British rule in India and other colonies was hardly democratic, regardless of who was actually ruling.

So, comparing all three we find that only 42 years separate the fall of the last French Monarch and the Fall of the last Qing Emperor, which is short enough for many people to be alive to witness both. England had been on a gradual trend (more or less) toward democracy as we know it today since before Queen Victoria's reign, and it would continue that trend after her death. To throw in one more example, the German Empire did not fall until 1918 and it took less than 30 years for Germany to be swept up by another totalitarian regime.

The complete dissolution of a monarchy in France did not occur until the late 19th century, and it did not occur in Germany until the early 20th, which is very similar to the timeline seen in China. Many other European countries retained their monarchies, and slowly transitioned to democracy over lengthy periods of time. Other European countries spent the better part of the 20th century being ruled by totalitarian governments. China's attempted transition to democracy was simply abrupt and unsuccessful, very similar to the French Revolution of 1789. So in short, the Enlightenment Era in Europe "demolished" very few monarchies. Depending on your interpretation, you could even argue that the Enlightenment didn't demolish a single one.

Source:

Holcombe, Charles. 2010. A History of East Asia: From the Origins of Civilization to the Twenty-First Century. First Edition edition. New York: Cambridge University Press.

EDIT: Fixed typo