I am seeking an effect for composites that creates a moving texture noise, such as oil painting. For reference, see the example below:
Please don't forget to demonstrate pictures, gifs, or upload blends files. to make it easy to understand.
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A cheap but not very controllable way could be to exploit the looks of a denoised render when there's too much noise. Basically, do all you can to make the denoiser do the worse possible result : lots of noise and too few pixels to work with.
Lower your rendering resolution, as well as your samples.
Set the denoiser to OptiX as it tends to perform worse than OpenImageDenoise at lower samples, you can even set the passes to "None" to "help" him doing a worse result.
It produces weird shapes. If your render is even more complex than mine with geometry everywhere, it should be even more visible. The first example you show really looks like what can happen with that method.
I was about to advise that answer, but it looks like textures are really broken now in Blender, so let's do even simpler:
Find a form of animated noise on the web, may it be a video or a GIF.
In the compositor, enable "use nodes" in the header. Drag and drop your noise file in there. Depending on the one you got, you can already multiply it with your render layer by pressing Shift A > Color > Mix, and plug in there your render layer in the first image, then your noise texture, set the mode to multiply and play with the factor.
You can also add a Color > Color Ramp to adapt the darkness of the noise, a Distort > Transform node to scale the noise up, and a Filter > Blur node to blur it up. Pretty much as follows: