This page talks about paths and mentions the ever-useful option to wiggle paths.
That's assuming your animation is created with shapes in AE. How are you creating your animation?
- The Great Szalam
(The 'Great' stands for 'Not So Great, in fact, Extremely Humble')
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Paul, I do not know if it is possible to achieve the same style just with After Effects.
I did a job recently that was a parody of Red Bull spots and got a very similar result to the original, but using Cinema 4D.
Take a look: