This is a somewhat difficult question to answer because of its breadth. Many different individuals and groups did many different things because the "white army" was not a single entity so much as a broad coalition of different groups united only by their opposition to the "red" bolsheviks.

Some fled the country after the war ended. There were significant Russian expatriate communities in China (harbin and shanghai) Europe (paris and berlin) and even in America after the revolution. There were organizations that supported "white" anti-communist causes among expatriate communities into the 30s. some of those ended up fighting in various anti-communist causes down the road.

Some were arrested and imprisoned and executed.

Some integrated back into society and became loyal to the communist government. Some fled into the woods and continued the fight as guerrilla movements until the 1930's.

This is not a comprehensive source by far but I highly recommend the book "The bloody white baron: the last Khan of mongolia" by James Palmer as an engaging story about one particular group of former white army officers who set themselves up in Mongolia.