family sharing

 If there are children under 13 years old, the new organizer needs to leave the existing family, set up the new family, and then the under 13 years old children can be transferred to the new family. Then disband the old family and join the new family.

Family Sharing -  Transfer a child to a different family group - Apple Support     

Family Sharing - Leave.          

The current organizer will need to stop Family Share. Then the new person will send out new invites. https://support.apple.com/en-us/HT201081