The origins of this move are ancient but were made into reality with the Glorious Revolution of 1688-9 in Great Britain. Prior to that time, every monarchy in Europe was absolutist, at least in principal, with the monarch having complete control of all aspects of government except, perhaps, the church. The Glorious Revolution installed the first constitutional monarch on a European throne. Now don't get me wrong, s/he still had a lot of power, but it was far more controlled by the English and Scottish Parliaments. In the subsequent years, British monarchical power declined to such a degree that in 1841, the young Queen Victoria wasn't even allowed to veto the choice of prime minister. In fact, Britain survived the waves of revolutions in the later centuries because the parliament already had control of the monarchy.
The rest of Europe was in a lot worse condition. The French king's heavy-handedness over the years finally resulted in the French Revolution, which saw the king turned into a constitutional monarch only to be deposed and beheaded a few years later. France later became a new monarchy under Napoleon who tried to create a more nationalistic state. He didn't fail in that goal, just in his tactics. When the old French monarchy returned in 1814, the people were not exactly as loyal as before to the king and deposed the monarchy again in the Revolution of 1830, resulting in a second constitutional monarch, Louis-Philippe. Unfortunately, the royal desire toward absolutism caused the new king's downfall and France overthrew yet another king in favor of a republic in 1848. Four years later, Napoleon's nephew, Napoleon III, took the imperial title and attempted to rule France like a proper constitutional monarch should, but he failed too, with Paris declaring The Commune and the Prussians marching in to abolish the last French monarchy. France's monarchical history ends in 1870.
Other states of Europe were also affected by these waves of revolutions and invasions. The Kingdom of the Netherlands was created after the Napoleonic Wars only to be divided in 1830 creating Belgium. Italy abolished all its hereditary monarchies save Savoy's to create the Kingdom of Italy. Soon after, Spain dissolved its own monarchy for a year to test republicanism. Portugal, after tough years and the secession of Brasil, finally overthrew its monarch in a republican-led coup in 1910. Meanwhile, Japan, marching across Korea, abolished the Korean Empire in 1910, while China overthrew its own 2000-year-old monarchy two years later. The world was in flux and the trend was toward communism, republicanism, or constitutional monarchy. Only the last choice allowed a monarch to stay on the throne, and most countries chose the other options.
The proverbial crap really hit the fan once World War I started, though. Queen Victoria had been busy in the previous century and was grandmother or in-law to the majority of important monarchs who participated in World War I. Somewhat oddly, Franz Ferdinand, Archduke of Austria, was not one of her kin but his death caused country after country to join in the war pitting Germany, Austria, the Ottoman Empire, and at times Italy against the United Kingdom, France, Russia, and at times Italy. Nicholas II of Russia, grandson-in-law to Victoria, was the first to go. His country, tired of centuries of oppression by the upper class, finally rose up and overthrew the dictatorial Romanovs to begin the decade-long process of creating the Soviet Union. With no reason to keep them, the Soviets murdered Nicholas, his wife, and his five children.
Germany soon suffered a similar, albeit less violent, fate. Germany was losing the war and the people were already getting angry. While the country had been united in 1871, the kaisers had allowed the rulers of the once-independent monarchies to retain their titles and status, delegating military and foreign affairs to the imperial government. This proved to be disasterous. Already a few monarchs had been deposed and the states had abandoned the German Empire. In November 1918, facing increased threats to his life and position, the German kaiser abdicated the German throne. Within two months, all other monarchs in Germany, about nineteen in total, abdicated their thrones or were forced off them. Kaiser Wilhelm II, grandson of Victoria, had to flee to the Netherlands where he lived until the middle years of World War II. Some of the other monarchs were allowed to stay but they were barred from participation in government or use of their hereditary titles or names.
Austria, Hungary, and Serbia all suffered the same fates. Austria & Hungary were united under the Habsburg-Lorraine family. Austria deposed their monarch and Hungary declared the throne vacant until the 1940s. Serbia became a much larger state but abandoned its former monarch.
* Mongolia - 1924: Fell to communist forces
* Spain - 1931: Fell to fascist and nationalistic forces (re-established in 1975)
* Albania - 1939: Italians dissolve monarchy
* Croatia - 1943: Italians abandon country
* Iceland - 1944: Seceded from Denmark and declared a republic
* Yugoslavia - 1945: Became communist republic
* Manchuria - 1945: Japanese puppet state, merged into China
* Italy - 1946: Monarchy dissolved after referendum
* Bulgaria - 1946: Soviets dissolve monarchy
* Romania - 1947: Fell to communist forces
* Ireland - 1949: Seceded from union with Britain
* Egypt - 1953: Revolution ended monarchy
* Vietnam - 1955: State partitioned and republic declared (following secession from France)
* Iraq - 1959: Monarchy overthrown in a coup
As you can see, the first half of the twentieth century saw the decline of much of the world's monarchies. In the subsequent half century, most monarchies in Africa too were dissolved. Now only a few remain and virtually all of them are constitutional, meaning the monarch is mostly a figurehead. Indeed, monarchs are not even allowed much power in order for their country to be a member of the European Union.
Dynasties will continue in perpetuity but their dynastic lands and titles change and end with time. The trend has been away from monarchism but it could someday return. France, Italy, Portugal and Germany all have monarchist movements to restore some form of monarchy and it has happened in some countries such as Spain. Perhaps the trend will change but for now, we must all see that the twentieth century witnessed the demise of the absolute monarchy.