Hum, There might be something wrong in the way they are rigged.
It could be a cyclic dependency, like an object that dictates how the curve should be shaped, but that object is parented to the curve.
Sadly the only way to be sure is to hunt down the issue, you can start by removing all the unrelated objects to simplify the problem, then desactivate some fuctionality or remove more objects until the problem disapear , once you find the culprit the solution in general is easier !
Did you tried to render without persistent data and motion blur ?
at least , removing persistent data won’t impact quality, only a bit more render time…