There were two problems:

1) When starting the mimic, we put up a bad frame because although we set the
   `begin` value of the animation, we were building using the current `value`,
   which hadn't been updated.

2) When stoping the mimic, we'd dirty a component during didUnmount, which
   wouldn't get cleaned until the next frame. Now we're sure to clean all the
   components before leaving flushBuild.