I don't know how to quantify the relative rareness of executing toppled monarchs, not having any sort of database of toppled monarchs to judge against. Without a comprehensive dataset to judge on it can be hard to get a grasp on, because the cases where monarchs were executed are particularly famous: Charles I of England, Louis XVI of France and Nicholas II of Russia.

But it's also quite common for toppled monarchs to go into exile. Sometimes this happens because the monarch sees the way the wind is blowing and flees; other times it's a sentence handing down by the invading army or revolution/coup that toppled the monarch in the first place.

In France, my specialty, Louis XVI was of course executed. But over the course of the 19th Century France had five more instances of monarchs being dethroned, and in every case they went into exile rather than being executed.

• In 1814, Napoleon abdicated after being defeated by invading Allied armies. The treaty by which Napoleon agreed to abdicate sentenced him to exile on the Mediterranean island of Elba, which was granted to Napoleon.
• In 1815, Napoleon escaped from exile and returned to power in the Hundred Days, until being defeated at Waterloo. He tried to flee the country for America but dawdled and was captured by the British, who, with the acceptance of the Allies, send him to St. Helena. (Some among the Allies, chiefly Prussians, wanted to execute Napoleon, but this was a minority opinion, and fortunately Napoleon wasn't captured by a Prussian contingent.)
• In 1830, Charles X abdicated as a result of the July Revolution. He went into exile in Britain (and later Austria). Charles left the country ahead of an angry mob, but the leaders of the revolution (including Charles' cousin, soon to be crowned as King Louis-Philippe) explicitly wanted Charles to survive and escape, and enabled this to happen.
• In 1848, Louis-Philippe was in turn forced to abdicate by a revolution. After abdicating he fled to England in disguise and remained there until his death.
• In 1870, Emperor Napoleon II was defeated and captured by invading Prussians. When they released him at the end of the war, Napoleon II chose to go into exile in England, where he remained until his death.

Similarly, while Charles I was beheaded by a revolutionary government, his son James II went into exile in France after the Glorious Revolution.

The final Austro-Hungarian emperor went into exile first in Switzerland and then in Portugal; he was not executed even though he tried twice to restore the monarchy and was captured by the Hungarian Republic.

Kaiser Wilhelm II fled to exile in the Netherlands after World War I; the Treaty of Versailles called for prosecuting Wilhelm for war crimes (British Prime Minister David Lloyd George called explicitly to "hang the Kaiser," but the neutral Netherlands refused to extradite Wilhelm, and other Allies, most notably U.S. President Woodrow Wilson, opposed Wilhelm's extradition and trial.)

In 1808 Napoleon I imprisoned Spanish King Ferdinand VII and forced him to abdicate, but kept him alive. Later Ferdinand was restored by Napoleon, only to be faced with a revolution in 1820. The rebels maintained the monarchy, with Ferdinand as a prisoner, and forced him to restore a liberal constitution. Despite having Ferdinand in their power the rebels never executed him. Later Spanish monarchs to be deposed all ended up in exile rather than executed.

In World War II, the Nazis occupied a number of European monarchies, including Belgium, the Netherlands, Denmark, Norway, Greece and Luxembourg. The kings of Belgium and Denmark remained under occupation (and were not formally dethroned by the Germans) while the others fled into exile.

As you can see from this brief survey, going into exile is a pretty common fate for monarchs who have suffered a successful revolution or foreign invasion. Royal executions certainly do happen, but at least in modern times it's probably fair to describe them as more the exception than the norm.

As to how the monarchs lived in exile, the key question was always how much support their host country gave to them. Sometimes their hosts would be generous, granting a pension and the use of a castle or estate; if the host government did not so, local elites sometimes did. (When the host country had diplomatic relations with the successor regime, they would often, for reasons of diplomatic nicety, prohibit their guest from openly proclaiming their claim from exile. Charles X of France was required to call himself the "Comte de Ponthieu" in his British exile; Louis-Philippe went by the "Comte de Neuilly.") Other times the exiled monarchs would be merely tolerated; Napoleon III had to sell jewels and properties to support himself.