Translate texture from outer bounds to center The motion I want is very close to water soaking in the sinkhole (just without spiral aspect).

I'd like to use Noise texture on a flat plane - it should move to the center and preserve pattern as much as possible, without morphing into "Saturn rings"
You could manipulate texture space coordinates. You would need to make one go from the center outwards and then animate a value added to it. Gradient texture could be used: