The nobility was not one big blob in the social hierarchy of the Ancien-Regime. This is why we have to distinguish the nobility from the aristocracy, especially when we talk about the 18th century. The difference seem very thin and hard to see. Being a noble means having a legal status, belonging to the second out of the three orders. An aristocrat on the other hand is, if we us the etymologic meaning, the leading class, the class in power ( aristos = the best/the most excellent and kratos=power ). So not all the nobles were at the top of the society and as a matter of fact many were extremely poor financially or of a little esteemed or simply not close to the power. The hierarchy in the nobility is very complicated, perhaps even more than the " rest " of the society. I say this because it is a difference important to make to understand the role of the nobility in the Revolution. Patrice Higonnet, who is teaching at Harvard, says ( see the first source below ) that this a difference that many Revolutionaries, like Jacques-René Hébert, did not make and as such for them, all the nobles were aristocrats, and so all were naturally against the Revolution and the Great-Nation that is France.

I will keep the term noble for the following answer.

Not all of them fled France. Many did not. We will see several examples :

The Revolution was a true seizure for the nobility. They took a heavy blow because the world they knew and were a part of was reshaped. Wether or not the nobility and the aristocracy were responsible for the revolution and wether or not they fared well during and after it is a huge debate, at least in French historiography, between historians like Philippe Bourdin, Guy Chaussinand-Nogaret, Bernard Bodinier or the late François Furet. All this to say that nobles took very different paths during the Revolution from instigating it, supporting it, fight it, living through it or simply giving all up.

There are those who as soon as the Revolution started emigrated in the hope to fight it or wait for it to go away. Among those there are many princes du sang ( Princes of the blood ) like the Bourbon princes ( the duke of Enghien and the princes of Condé and Conti ). Not all fled of course, Louis-Philippe d'Orléans, prince of the blood, supported the Revolutionary army and voted for Louis XVI's death. The King's brothers, the counts of Provence and Artois ( future Louis XVIII and Charles X ) did not last long before living France. They were very much opposed to any transgression of the King's divine right to rule, just like they were opposed to Louis XVI's reforms attempts to re adjust France's financial situation.

Just like the nobility is not one solid group, the Revolution is not one big event. I will let someone else, if you wish, go into the political details of the Revolution but not everyone supported the Revolution all the way through. The Revolutionaries of 1789 are not always the same as the ones of post-Louis XVI. François-René, viscount Chateaubriand is an example of a ( minor ) noble sensible, if not supportive, of the Revolutionary ideas but who soon turned to hate the blood-shed the Revolution turned in ( as he said, when he saw the heads of fellow nobles on spikes, his disgust for the Revolutionaries grew stronger ) and he decided to leave France for the Americas.

Many of the most prominent Revolutionaries were from the bourgeoisie but a high/important/rich bourgeoisie who were, before the Revolution, often close to buy a fief and/or an office which would give them a title. Many were already nobles, sometimes from very old families. The percentage of noble/bourgeois/other was not the same in every faction. Also, the old nobility ( noblesse d'épée, the military nobility ) did not react the same as the nobility of the robe who were generally relatively new nobles families and had offices in the administration, finances,...Etc Louis-Michel Lepeletier, from the nobility of the robe turned to be one of the most virulent advocate of the people. His ideas were among the most radicals ( suppression of noble sounding surnames, like his own or the fact that children should not belong to their parents but to the state in theory ). He defended and voted for the death of Louis XVI but was murdered soon after by a fellow ( but very minor ) noble, Philippe Marie de Pâris, an ex-guard of the king who had swore to avenge the king.

Marie de Pâris is representative of a certain category of noble. He did not act as much to protect himself and his situation as a nobleman as he acted to protect the monarch. Many of the nobles who emigrated did it in order to reinstate the old order and not necessarily to protect and/or avenge the King, whom they did not necessarily liked because of his attempted reforms. The Army of the Princes ( or Army of the Emigrés ) was composed of many nobles, from big or lower status like Louis de la Forest Divonne or all the male members of the very old d'Allonville family. On the other hand, nobles also served the Revolutionary armies, not necessarily all the time though like the famous count of Rochambeau ( the veteran of Yorktown and the American War of Independence ) or Napoléon Bonaparte. The Bonaparte were a noble family of Ancien Régime since 1769.

There also was the case of nobles fighting in France, mostly with Catholic Army around 1793. Several nobles, often from minor families who had returned to France or never left. Some names are quite famous like La Rochejacquelin, Charette de la Contries ( two of the heroes ), Gigost d'Elbée, de Bonchamps, ...Etc Or in the army started by the woodcutter Jean Chouan ( the Chouannerie ) like the marquis of La Rouërie, Marie-Paul Scépeaux de Bois-Guignot or Picquet du Boisguy.

I gave you a lot of examples to explain that there is not really a " pattern " to explain the reaction and the attitude of the nobility toward the Revolution. Many supported it, many emigrated, many fought back but you often have to do case by case as their opinion changed and so did their actions. There are studies giving percentages of how many nobles from X French region fled or joined the Army of the Princes but so far I havent't read, at least in French, a global study giving a proper number about how many emigrated. Even if there is one, it would participate to make the nobility like a solid group of people, living and thinking with the same standards and ideas. It's easy to calculate how many princes of the blood did what but much harder to do it for the entire nobility, especially when you research the lowest of the low with the nobility who could not dress better than a peasant. For example, Philippe Bourdin in les nobles Français dans l'Europe de la Révolution says that around 69% of the nobles from the Puy-de-Dôme department fled France while only 10% from the Essonne ( close to Paris! ) did. But there are a lot of factors to take into accounts like who were these nobles in general. In the West near the Vendée and the Chouans, many of the peasantry went looking for nobles, who had experience as military leaders when they launched their counter-revolutionary war. This is how La Rochejacquelin found himself at the head of a Vendean army.

So to understand the decision of the nobles, we have to see who they were ( rich ? poor ? working for the King ? Sympathizers of the Revolution ? ) just like we have to see how the population around them was and the general situation of the region/location they were in.

Another good example is the insurrection of Lyon in 1793 against the National Convention. Now, despite what the English Wikipedia says about it, the uprising of the city was not a counter-revolutionary movement but one against the Convention. Most of the leaders of Lyon and the Sans-Culottes there had proclaimed their attachement to Republican ideas and were actually extreme revolutionaries. Yet at the siege of Lyon, which put an end to the revolt, two nobles lead the opposing armies. Kellerman for the Convention army and Perrin de Précy a royalist who still lead the Army of Lyon.

At the end, the nobility did survive the Revolution. Their world however was definitely over and the different values and traditions forever changed. Napoléon, the birth of the imperial nobility, the choices nobles made during the Revolution, ...Etc provoked a deep identity crisis among the nobility. They had to try to rebuild their world in a society that, if they were still very important, did not need them as much as it used to ( link to a previous answer I made ). As I said, difficult to say what the nobility as a whole ( since there is no such thing as the nobility in my opinion ) did. Even if a study says that ( for example ) 80% of the nobility did X or Y, we almost have to rely on case by case research.

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Sources :

Higonnet Patrice, « Philippe Bourdin (ÉD.), Les nobles français dans l'Europe de la Révolution, Rennes/Clermont-Ferrand, Presses universitaires de Rennes/Presses universitaires Blaise-Pascal, 2010, 600 p., ISBN 978-2-84516 », Revue d’histoire moderne et contemporaine 1/2012 (n° 59-1) , p. 175-176 Online

Dictionnaire critique de la Révolution française by Mona Ozouf and François Furet

Dictionnaire historique de la Révolution française directed by Albert Soboul.

La chouannerie et les guerres de Vendée by Nathalie Meyer-Sablé and Christian Le Corre

Nobility in France was actually much larger than you would expect, somewhere less than ten percent of the total French population were nobility to some extent. As a result of this, the French Revolution caused a massive upheaval but oddly enough this didn't cause everyone to go running.

This is the difference between petite noblesse (like the young nobleman from Annoux that I will speak about) and the Duc d'Orleans that ended up with his head chopped off. As there is a great different amount of nobles, depending on how angry the people were, how the noble lived, and how conservative they were; there was a chance that they would either leave or stay depending on how fearful of the Revolution they were.

However there were some, usually this petite noblesse or minor nobility that stayed and fought for the French Republic. One fine example is possible one of the greatest commanders of the era, Louis-Nicholas Davout. Davout was a minor noble from Annoux that was serving in the Royal-Champagne Cavalry Regiment as a sous-lieutenant. However upon hearing that the Revolution was happening he rode over with a sergeant and declared his loyalty to the new Revolutionary government, which got him in jail for leaving his post. He would continue to serve in the French army, despite being jailed again and having his rank revoked, he would enlist as a private and make his way back up until he was Napoleon's youngest Marshal.

The nobility was not a monolithic block that did a certain thing, many people stayed to take advantage of the merit based promotion system that would favor the skilled, such as Davout and a young Corsican, but others would take advantage of their position and help move France forward like Talleyrand, who would serve as Minister of Foreign Affairs throughout the Revolution and the Empire.

The nobility that did leave did either one of two things, they either hid in other courts or actively participated in the counter-revolution, trying to garner support from foreign governments to end the Revolution.

/u/OakheartIX and /u/DonaldFDraper , thanks for the answers!