I have Kids Unlimited and each of my kids has their OWN Fire tablet. Problem is that their internal storage is being constantly consumed by the fact that their siblings profile are being automatically downloaded onto all the devices. They do not need to share their device, so it is pointless that this is occurring and just caused memory consumption headaches all the time.
At this time there's unfortunately no way to delete the other child profiles off the other devices or to only download the things you're sharing to specific devices. I would definitely recommend you leave feedback on that though!