Jake, thanks for the encouragement. I'm sort of stumbling around in the dark, but Cheetah is definitely great to work with.....
OK, so here are multiple morph objects. I started with a sphere and tube as parent and child, then duplicated that three times before messing with the motion. (After some trial and error, it looks like the rotational paths are sine waves, each offset from the other.)
Then I duplicated the tubes, scaled them up and flattened them, positioning them in the object browser at the same level as first tubes. That way, their rotational motion is still controlled by the position of the spheres. With straight segments and obvious breaks, the motion still looked jerky.
Then I took one of the enlarged and flattened tubes, duplicated it and bent it into a gentle curve. I used the original as the morph base and the curved tube as the morph target, and then dropped the resulting morph object back into the tree and animated the morph path. To my surprise, that worked well (and negative values worked fine for the reverse curve--thanks Hiroto-san!), so I repeated the process three times for the other segements.
Finally, I tapered the last segement and stretched the innermost tube (yellow tail) just to emphasize the action.
The segments aren't joined perfectly, but with some scales or patterns or feathers on it, I'm not sure that would ever be noticeable.