The Holy Spirit is pictured in the Bible as descending like a dove, in the accounts of Jesus’ baptism (Matthew 3:16; Mark 1:10; Luke 3:22; John 1:32).
But there is nowhere in the Bible where human believers are pictured as flying like doves. Well, there is Psalm 55:6, but this is an unfulfilled wish rather than a God-given picture. Humans are likened to doves in a few other places in the Old Testament, especially in the Song of Songs, and in Matthew 10:16, but the point of comparison is never flight – except possibly in Hosea 11:11. That is to say, it is nowhere suggested in the Bible that humans will or should fly like doves.
But what we do find several times in the Bible is a promise that believers will fly or soar like eagles. See for example Exodus 19:4 (the Exodus compared with the flight of an eagle); Deuteronomy 32:11 (Israel as a young eagle being taught to fly); Isaiah 40:31 (“those who hope in the LORD … will soar on wings like eagles”); and Revelation 12:13 (the woman, symbolising Israel or the church, is given the wings of an eagle).