I'm going to begin by asking a slightly different question, if you don't mind, but I think it will allow us to better understand what happened. I am going to ask instead: "Why did the Allies send men and supplies to Russia in the first place, and how important was the Allied intervention to the outcome of the Russian Civil War?"
The answer to that question is that the Allied intervention was about winning the First World War, only accidentally ended up getting entangled in the civil war, and was not very important to determining who won.
Why were the Allies in Russia in the first place?
In the aftermath of the civil war, the Bolsheviks portrayed the Allied intervention in the war as a concentrated effort by world capitalism to smash the Red revolution in the bud. This is still probably the dominant interpretation most people think of when they think of the Russian Civil War, however it is not what actually happened at all.
Britain, France, Japan, the US, and the other Allied powers only ever half-heartedly supported the Whites. Japan, for example, sent by far the most men and material to Russia between 1918 and 1923, over 70,000 troops, but they never advanced at all beyond the Russian Pacific Coast. Their sole intention was to strip Russia of land and resources while the nation was in turmoil. The "White" warlords that Japan sponsored like Ungern-Sternberg were brigands and stooges; their miserable example probably did more to hurt the anti-Bolshevik movement than help it.
The other Allied nations provided a more help to the Whites, but not enough to matter. What Britain and France wanted when they sent troops to Russia was not to help the Whites win the Civil War. They did not particularly care to get involved in internal Russian politics whatsoever. What they wanted to do was maintain an Eastern Front against the Central Powers of World War I (which was still ongoing) by sending a nucleus of Entente troops around which the demoralized Russian could reform. Whether those Russians were Bolshevik, anti-Bolshevik, SR, Kadet, whatever, did not much matter. All that mattered to the Allies in 1918 was that the Russians kept fighting against the Germans. Before the Treaty of Brest-Litovsk was signed, the Allies were perfectly happy to work with the Bolsheviks, too. The first Allied troops to arrive in Russia was a contingent of Royal Marines who landed at Murmansk in March 1918, with Lenin's direct approval. The French military preferred to work with the Reds, in fact, as they seemed better equipped and motivated than the little groups of loyalist Russians organizing on the empire's fringes.
So how did the Allies end up fighting the Reds?
But of course the Allies did end up supporting the White governments like those of Kolchak and Denikin, so how did that happen? Again, it was never part of an organized plan. Lenin pissed off the Allies by defaulting on Russia's massive debts, publishing the Entente's secret diplomacy, and, worst of all, making peace with the Central Powers, allowing Germany to swallow up Russia's food resources as well as great quantities of war material it had been sent by Britain and France, and to transport her troops from the Eastern Front to the Western Front. Needless to say, this did not make Lenin any friends in the West.
However, Allied troops still did not go to Russia with the intention of replacing the Bolshevik Government. They went there to help the Czech Legion, (which /u/Georgy_K_Zhukov has written about in great detail,) a unit of the Russian military that had pledged loyalty to the Allies and subsequently fallen afoul of the Bolsheviks while it was leaving Russia. British, Japanese, and American troops landed in Russia to help get the Czech soldiers out.
From there, the Allies sort of stumbled into a war with the Bolsheviks. None of this was intentional on the part of their governments. Britain's cabinet and Prime Minister, David Lloyd George, did not want a war with Lenin. But many of the military officers sent to Russia and ordered to fight the Germans did not make any differentiation between the Germans and the Reds, who they tended to see as enemy agents in German pay. The latent anti-Bolshevism of the British officer corps ended up swaying these military missions towards conflict with the Bolsheviks, such as in Archangel. There an inter-Allied force had propped up a socialist Russian government with the aim of fighting the Germans and aiding the Czechs. The force was commanded, however, by a conservative British general named Poole, who soon launched a full blown war on nearby Red forces without any instruction from his government.
From there the Allied intervention in Russia took on a life of its own. Other British generals found their own anti-Bolshevik regimes to support. The obvious example is in Siberia, where Britain's military observer, Sir Alfred Knox, developed a liking for a Russian naval officer named Alexander Kolchak, and probably aided - perhaps even formulated, I think - a coup where the passionately anti-Bolshevik Kolchak replaced the more moderate Siberian government. Britain began sending Kolchak aid, entirely at Knox's discretion. In London only a few conservatives like Churchill and Lord Curzon supported the Whites, and they were constantly overruled by Lloyd George and the other members of the War Cabinet. But because the government never actually developed a coherent plan for Russia, it allowed unchecked "men-on-the-spot" like Knox and Poole to formulate government policy, and drag more and more British men and supplies into the civil war.
So why did the Whites still lose?
Even the support the Allies did sent at the behest of these men-on-the spot did not make a great difference, however. For one thing, to use Kolchak as another example, corrupt officials embezzled or stole much of the money and supplies that Britain sent to Russia. It was not uncommon in Siberia to see generals' girlfriends wearing British army greatcoats while the White troops in the front-lines froze to death without winter clothes. And even the quantities of supplies that the British did get to the front-line could not really make a difference. At the height of Allied support in late spring 1919, the Whites were at their high-water mark: they had Moscow surrounded on three sides by Kolchak's forces on the Volga, Denikin's Volunteer Army surging northwards from the Don, and another White army led by General Yudenich on Petrograd's doorstep. But the Reds had an enormous advantage in men and material, and within three weeks they had sent the Whites packing in all directions.
As it became clear to men like Knox that they had backed losers, they abruptly ended their support for the Whites. When the generals gave up their personal crusade, there was no impetus to keep Allied supplies flowing to the Whites. Almost no one in Britain understood why they were sending men and money to Russia in the first place, and many, like Lloyd George, had long since wondered if Britain would probably be better off with Lenin in charge of Russia rather than an incompetent warlord like Kolchak.
As for the case of the other Allies - they did not do much. Japan, as I said, wanted to steal Russian property, not aid the Whites. The Americans did not want to be there at all and left as soon as they could, and France's involvement did not go far beyond a cynical, somewhat pathetic attempt to create a puppet state in Ukraine to make back the money that had been lost when Lenin repudiated his debts.
Tl;dr
The Allied intervention in Russia was a half-hearted, unplanned venture which was part of the First World War and only ended up stumbling into the Russian Civil War because a number of British generals "went rogue". Even when the Allies did provide aid to the Whites, the lack of clear goals and strategy limited its worth, and White corruption and incompetence further squandered the help. In the long-run, the odds of the Russian Civil War were so stacked in the Reds' favor that the Allied intervention could do no more than prolong a bloody civil war.
Apologies for a long answer; I find it is difficult to really talk about the Allied intervention in Russia without first addressing many of the assumptions we have about it. The intervention needs to be clearly situated in the context of World War I and the Czech Legion to show why it was not really about helping the Whites against the Reds. You might also like to check out my answer about America's role in the intervention, which goes into some more detail in the US's case.
Sources
The link between British officers in Russia and the imperial tradition of men-on-the-spot is what I wrote my undergraduate thesis about, strongly inspired by Brock Milman's article "The Problem with Generals: Military Observers and the Origins of Intervention in the Russia and Persia, 1917-18." Journal of Contemporary History 33 (1998): 291-320.
For more general information on the Allied intervention in Russia, see also:
John Silverlight's The Victor's Dilemma: Allied Intervention in the Russian Civil War, 1970
George Kennan's The Decision to Intervene. Vol. II of Soviet-American Relations, 1917-1920, 1957
Jonathon Smele's Civil War in Siberia: The Anti-Bolshevik Government of Admiral Kolchak, 1918-1920, 1996
and George Ullman's Anglo-Soviet Relations 1917-1921, Vols. I and II.