The allegiance of Muslims during the Russian Civil War was complicated and depended on the time and place. In the Tatar and Bashkir regions, some inhabitants initially joined the Committee of Members of the Constituent Assembly, or Komuch, a government established at Samara in June 1918 by (mostly) Socialist-Revolutionaries opposed to Bolshevik rule, with help from the Czech legion. The Komuch was basically an attempt to revive the elected Constituent Assembly that the Bolsheviks dissolved in January, but it never really gained mass support or developed much of an army and after Samara fell to the Reds in October the Komuch was effectively subsumed into the wider White movement of Admiral Kolchak. Many Tatars and Bashkirs continued to fight for the Whites for a time, but they became disillusioned with the rigid refusal of the White leadership to recognize their national aspirations and began to see the Reds as a better bet. Zeki Velidi Togan, the most prominent Bashkir leader, defected to the Reds in March of 1919 and set up an autonomous Bashkir Republic with their support, headed by Bashkir Communists. Unfortunately the Reds soon decided that the industries of the region were too valuable to be left in the hands of the Bashkirs and reversed their support for Bashkir autonomy, decreeing in May 1920 that the Republic was under Soviet control. This sparked a rebellion against them by the Bashkirs. As for the Tatars, they initially chafed under Red rule (under which most of them had fallen by the summer of 1919), which failed to respect their customs and brought with it the Red Terror, but relations improved with the establishment of the Tatar Autonomous Republic in May 1920. While the "republic" did not actually involve much political autonomy, the Reds had recognized that repression would win them few supporters among the Tatars and switched to a stance of support for Tatar culture. The use of the Tatar language was expanded in schools and administration and Tatars were included in political institutions. This policy was supported by the Tatar leader and intellectual Mirsaid Sultan-Galiev, a Bolshevik since 1917. He was a believer in "Muslim communism", a kind of blend of Marxism and Islamic tradition, and he even led an independent Muslim Communist Party and Muslim Red Army units during the Civil War. The Reds allowed this, since the military situation meant they needed all the support they could get, but eventually decided that Sultan-Galiev's emphasis on national liberation went too far and he was booted out of the party in 1923 (though the policy of cultural if not political autonomy for the Tatars continued until the late 1920s).

In Central Asia the story was similar to that of the Tatars, as the Muslims of the Turkic peoples initially resisted the Reds but gradually came around to them. A group called the Basmachis rebelled against the Turkestan Soviet Republic established in the region in early 1918, angered by the forced requisitions and labour conscription policy of that body. Starting in early 1920 the Reds changed their tune and began to allow bazaars to open, reduced requisitions, and even restored Islamic law for some matters. This took the wind out of the Basmachi's sails and the rebellion petered out by 1923. The Soviet regime threw its support behind cultural initiatives in Central Asia (especially language policy) and by 1921 the majority of Soviet delegates in the region were Muslim, indicating that most local Muslims had come to accept the Revolution and the Bolshevik government (with varying levels of enthusiasm, of course).

As for the Caucasus, Azerbaijan was an independent nation from May 1918 to April 1920, but during this time the Reds built up support among the Muslims by leveraging economic discontent with the government. By Febuary 1920 there were some 4000 Bolsheviks in Baku and Tiflis, and the Red Army launched an invasion with support from the Turks soon after. There was little resistance by either the army or the people, a few uprisings were quickly crushed, and Azerbaijan was in Soviet hands by the end of April. I don't know much about specifically Muslim support or opposition to the Reds in Armenia or Georgia, but the conquest of these nations followed similar patterns to Azerbaijan: a brief period of independence, undermined by Bolshevik internal agitation, followed by a successful invasion with Turkish support. By the spring of 1921 the whole Caucasus region was under Soviet control. In conclusion, the Muslims of Russia during the Civil War variously supported the Reds, the Whites, and their own national movements, or a mix of these, but the majority of them gradually came to support the Reds, or at least accept their victory, over time. The Red change of policy with regards to national cultures (becoming much more liberal), a change that took place throughout Russia and called korenizatsiia (indigenization), was a key factor in this.

Source: A People's Tragedy, Orlando Figes