You can actually extract flapping motion, save it as Motion+ and apply separately to wings whenever you need it (even if wings are not attached to avatar).
I am not sure about "Shining Wings", but looks like they have the same principal as default IC6 wings.
While Wings are attached to any avatar, apply some flying motion (or use Move command from menu).
Next, Detach Wings from Avatar, bring Wings to the timeline, enable Animation track, right mouse click on that track and click Split.
That would separate Wings flapping motion from the avatar motion.
Find the initial Wings pose frame which is closest to the end of the track. Break the clip at that frame and delete right portion of it.
Now that you have perfectly matching flapping motion, you can extend it as many times as you'd like and save the clip.
Wings could be attached to anything now (props, non-human, animals, etc) and have an independent flapping motion applied.