Alter legacy animation at run-time

I have a scene with some characters in it and their animations are all legacy. Suppose I have a character walking around in that scene and the walking animation includes a little bit of arm movement. What I'd like to do is alter that character's arms in that animation to swing ever so slightly differently as he's walking around. Essentially add something like Perlin noise to it in order to save me the hassle of having to make a lot of animations that are nearly the same.

For the sake of argument, let's assume I can magically get the position/rotation that I might need for the appropriate bones. How would I go about altering the arms? Can you simply change the transform of the appropriate bone in the LateUpdate function? Altering anything in the Update function seems to get overridden. I don't want to stop the animation and deal with everything, just perturb the arms a little. Any suggestions would be welcome! Thanks.