Your question presupposes something that was rarely the case, namely, that a minor Russian noble would flee Russia for Europe with nothing but the clothes on their back. The most typical approach for nobility of all stripes was simply to pilfer whatever worldly holding had been bequeathed them and make for the hills.

You see this kind of behavior outside the strictly speaking "noble" class as well-- the White general Grigory Mikhaylovich Semyonov plundered about 36 tons of Romanov gold while it was en route to Vladivostok on the Russian Pacific coast where it was to act as a down payment for additional war materiel in support of the White Russian war effort^([1]). Semyonov used this gold in service of his ultimately failed attempts at creating his own sort of "trans-Mongolian" nation-state that would serve as a buffer between Japan (who more-or-less supported his intentions) and whatever state fate deigned to grant victory in Russia after the conclusion of the Russian Civil War.

Ultimately, the Bolsheviks were able to reclaim a large deal of that gold, but Semyonov escaped to a series of overseas hideaways until returning to Japanese-occupied Manchuria, being captured by Soviet paratroopers during World War II, tried as a war criminal (which he was), and finally being hanged. Literal tens of tons (i.e. billions of dollars) of Romanov gold is still unaccounted for to this day, underlining just how common the smash-and-grab technique was in Russia during this time^([2]). But I digress.

So, okay, let me try to answer your question as it is asked: "what happens to you if you're a minor Russian noble with only the clothes on your back and you want to flee Russia?"

Well, the first problem you're going to run into is getting to one place or another with no money, which I don't just say out of tautology. You may think I'm splitting hairs a bit here, because I suppose you're more interested in the substantive result of what happens when you (the noble) actually end up where you are going but in fact, where you might be able to go could very much be affected by a lack of capital upfront.

This map really nicely illustrates the Russian western front (or European eastern front if you like) during World War I. To avoid any additional hairsplitting, let's use that big yellow line labeled Front at the Time of the Russian Revolution 1917 as our tentative "border of Russia" for this answer. You can see here that essentially the most direct routes to eastern Europe, and indeed western Europe (your goal) as well, are going to be kind of tough going because they are mostly infested with the Hun menace. Thus, the time-tested method used by many fleeing White Russian emigres of simply bribing German border patrols into allowing them passage is not available to you, as a penniless Russian minor noble.

Enter Placement Familial, the French non-profit agency whose sole purpose was helping Russian (as well as Polish, Ukrainian, Belorussian, and other) orphans fleeing persecution of some kind on the territories of the former Russian Empire by setting them up with a French foster family or their own home were they of an appropriate age^([3]). Now obviously, you can fill in the blanks a little bit here to reckon that you'd have to be of an age where claiming orphanhood was reasonably believable, but what kind of documentation could you really be expected to provide if you just showed up from the war-torn steppe with only the clothes on your back? And more importantly, how can you get to France in the first place?

There were large-scale mass emigration movements whereby Russians of all classes of birth fled to Constantinople before sailing onward to Europe via the Mediterranean Sea, effectively bypassing that ominous yellow line in that map I included above, but many also went north into Finland and then onward into the Scandinavian (and more importantly, neutral/non-aligned) north of Europe. From there, they could make their way to Great Britain or France, and then onward to America if desired. Russian emigre communities provided a number of services intended to help refugees (which is essentially what you amount to as a noble with no money in a court that no longer exists) escape the horrendous conditions that were present in the former territories of the Russian Empire as you'll see below.

Likewise, many Russians (and Poles, Ukrainians, Belorussians, etc.) fled into Bukovina and Moldova while it was still possible. Ultimately their goal was the historically contentious "Slavic" regions of what would become Yugoslavia (translatable as Southern Slavic Land). This is where the hairsplitting comes into play again though because as that map above shows, the Austro-German advance had sealed this passage off by 1918. Again, bribing border guards was still very much something that happened, but for the sake of your question-- that's not an available option for you.

Thankfully, the ever-generous French Assemblée Nationale went so far as to endow several hundred 450,000 franc scholarships for promising emigre students to continue their studies in France (roughly $45,000 in 2019 dollars). Acceptance means your entire immediate family is eligible for relocation as well. Furthermore, your school-age emigre offspring will be admitted to government-funded French schools with no questions asked-- under the exact same admission requirements as a citizen of France.

In the early twenties, before the Bolsheviks had officially won the war (although by this point, it was quite clearly just a matter of time), Britain established the Russian Refugee Aid Society^([4]) and a loose confederation of German-Russian-American emigres established the American Volga Relief Society^([5]). In conjunction with the more well-known Red Cross, these organization worked to provide food, shelter, clothing, resources, and other material support for the millions of victims of the eastern front bloodshed and internecine civil war. This material aid is another reason that Russian emigres were rarely left to fend entirely for themselves.

If you managed to keep yourself alive until 1925-- about three years after the Red victory in the Russian Civil War-- the League of Nations (est. 1920) finally managed to get through the red tape and (in conjunction with the International Labor Organization-- which was eventually subordinated to the United Nations upon the latter's formation many years later) declare the world community responsible for the so-called "refugee problem" caused by the hugely destructive wars that had torn eastern Europe (and indeed most of western Europe as well) asunder. In fact, the legal definition of a "refugee" (and their associated rights) originates from this period according to the American Association for the Advancement of Sciences^([6]). That's something new I learned while writing this answer.

This "refugee problem" amounted to several hundred thousand Russian refugees (which includes our broke emigre noble in the context of your question) of which nearly 100,000 were brought into work placement programs in America, Britain, and France in one of the first real shows of international solidarity that the League of Nations managed to put together.

All this shows that, in fact, your options were of course limited, but absolutely not totally destitute if you found yourself on the wrong side of the Bolsheviks during the course or at the conclusion of the Russian Civil War. All of the major western powers provided not insignificant amounts of capital investment toward refugee programs intended to settle and help struggling Russian refugees as well as other more restorative initiatives such as the aforementioned job placement program.

Further Reading

[1]: Wonnacott, Collin James; Reassessing Russian Warlordism: The Case for a New Paradigm; 2016 (see pp. 50-55)

[2]: Liesowska, Anna; Fascinating new clues emerge on lost tsarist gold and diamonds worth an untold fortune; 2017

[3]: Hassel, James; Russian Refugees in France and the United States between the World Wars; 1991 (see pp. 29-30)

[4]: Gatrell, P.; Refugee History and Refugees in Russia During and After the First World War; 2017 (see pp. 503)

[5]: American Historical Society of Germans from Russia; Volga Relief Society; 2019

[6]: University of Helsinki; The League of Nations Created the Definition of Refugee; 2013

The French fashion industry was actually an employer of Russian émigrés!

At the beginning of the twentieth century, Russian embroidery and lace were quite fashionable on white shirtwaists, as they'd gone from being largely sportswear to something more versatile. They also began to take some inspiration from the cut of the Cossacks' asymmetrically buttoning, loose, high-collared coat, although it's not until the early 1910s that Russian and Cossack blouses really hit the market, with the latter cut straight across at the bottom and the former a cutaway jacket. (Here's is an example of Russian blouses/blouse effects in girls' dress, from The Delineator, 1913 - you can see very similar influences in a lot of women's clothing throughout the magazine, with bodices buttoning far to one side all over the place. The traditional kokoshnik headdress would become a high-end staple in the early 1920s. The "Ballets Russes" was an avant-garde ballet company based in Paris that collaborated with artists and fashion designers of the 1910s to create fantastic displays, sometimes incorporating colors, fabrics, cuts, and patterns seen in Russian folk dress, particularly in productions designed by Leon Bakst, Natalia Goncharova, or Mikhail Larionov. Goncharova also collaborated with the non-costuming couturier Nadezhda Lamanova, who had locations in both Paris and Moscow.

As a result, Russianness had a certain cachet that could be used as a selling point for a Russian's product or as a reason to hire them, and an understanding of Russian styles could be very helpful. Those who got to Paris with more money left started businesses: Atelier Kitmir was a Russian embroidery workshop started by Grand Duchess Maria Pavlovna that was patronized by couture houses; Maison Irfé was opened by Prince Felix Yusupov and his wife, Princess Irina, and traded as much on their good looks, notoriety, and close relations to the former Tsar as on the actual clothing they sold. Those with less money might work for them as dressmakers or embroiderers, needlework having been a normal accomplishment for noblewomen, or as fashion models, as they typically already had been trained in poise and an elegant bearing.

(This is really only a portion of the picture - for one thing, the vast majority of Russian émigrés who went into fashion were women, and I don't really know what their male counterparts did. But it is relevant and hopefully interesting!)