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.