How can I drag and drop a whole container?

I've been experimenting with a random character creator. I have sprites for different elements like head, eyes, nose, etc. all in one container. Right now I have a function that creates a new unique character every time it runs.

I want to be able to drag and drop the characters that are created, but the drap and drop behavior apparently does not affect the whole container.

How would I go about doing this?

You could probably figure it out with families. I've never experimented with this, so I have no idea. I hope my idea was helpful!

Make all the parts in the container children of a single parent ( a parent that has drag and drop).