I needed to omit the normal Callout.close() function and instead use removeFromParentWithEffect() first and then close.
This is currently your best option. I'd like to add a closeEffect, similar to the one in the new Toast component, but I haven't gotten around to it yet.
There is an addedEffect() but no removedEffect().
There will be no removedEffect in Feathers because there's no way play an effect when removing a Feathers component from a parent that is a regular Starling display object. Starling simply removes the child immediately. That's why removeFromParentWithEffect() exists.
If you ever used Flex, you may know that its components had a removedEffect. However, Flex had stricter rules about how its components could be used. Flex components could only be added as children to other Flex components, so it was able to override removeChild() and do something before calling super.removeChild(). Since Feathers components can be used anywhere in a Starling project, they need to be able to handle effects and things like that entirely by themselves.