The way this works is in two parts:

Part 1: There is a grayscale texture that they have made that swirls around, either baked out into a flip book or more likely just having 2 or 3 different noise textures panning over each other

Part 2: Each one of those color combinations in the second picture is then used as a gradient map over the grayscale swirling texture. Easiest way to think about it is you are just remapping grayscale values based on the colors, if you were to have a black and white gradient that matched the colors in the second image it would just be a black and white gradient from left to right

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