Watch the tutorial here: https://youtu.be/459FjmLx-XM
How to animate a gradient:
Create a new solid layer (shortcut is cmd+y) an give it a fill. (This just makes it easier to see and to change the color in the future.)
Now draw a mask around your solid. Whatever shape you desire. Now open your little dropdown menu for the mask, use mask feather, and adjust the feathering to be more fuzzy.
Add a "turbulent displace" effect. You'll find something called "evolution." Hold the option key and click on that to bring up the expression window for this. When I say expression, I literally mean expression. It's like a tiny excel sheet. For sake of tutorials, type "time*200." It will now be continuously animating.
You can then adjust the "size" to accomplish the look you're going for.
Now, copy and paste your masked solid layer and change the color. Adjust the points so the two colors flow instead of sitting on top of each other.
If you don't like that these two are warping at the same speed and pattern, go into evolution > evolution options > random seed. Alter the random seed to create a bit of variation.
Continue duplicating and altering layers as much as you want. You can also apply the "gradient ramp" effect to any of your layers... and now you have a gradient inside your gradient.
This is the basic approach to animating gradients, but the tutorial goes into more detail in case you need to create any extra movement instead of just a simple animation. Reference that for anything complicated.