How to create swirl effect

[#1]

I'm trying to mimic the swirl effect on the color gradient at 0:13 in this video. 

https://vimeo.com/278048098

I would like to create something similar for what I'm working on right now. 

https://www.youtube.com/watch?v=iyjo8J4a9vo

I tried applying a four color gradient and turbulent displacment effect. That, obviously, didn't work. 

[#2]

1. Create a gradient on a shape layer rectangle and then distort the layer using your choice of Distort Effects like Mesh Warp
2. Add another shape layer the size and shape you want your object and set it as a track matte for the Gradient layer

Pre-compose the pair and animate position and composite with other layers.

[#3]

Mine doesn't look like that AT ALL.  This is what I see.

[#4]

Rick has a good idea. I'll add another – use the Mask Feather Tool along with animating the Mask Path and Layer>Transform>Position/Rotation props.