There is no need to move any content. Each family member continues to use their own Apple ID for their iCloud storage but the billing is to the family plan. More information in Share an iCloud storage plan with your family - Apple Support including the following:
If you already had an iCloud storage plan
When you share your iCloud storage, any family members using the free 5GB plan are added to your family plan automatically.
If a family member is already paying for their own iCloud storage plan, they can choose to switch to your plan or keep their own and still remain part of the family. When they switch to the shared family plan, they get a refund for the remainder of their personal plan. They can't keep their current plan and use the shared family plan at the same time.