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To change the owner of a folder, first invite the user you wish to transfer the folder to as a collaborator on the folder. 
curl -i -X POST "https://api.box.com/2.0/collaborations" \
     -H "authorization: Bearer <ACCESS_TOKEN>" \
     -H "content-type: application/json" \
     -d '{
       "item": {
         "type": "file",
         "id": "11446498"
       },
       "accessible_by": {
         "type": "user",
         "login": "[email protected]"
       },
       "role": "editor"
     }'
Then, update the created collaboration by changing the role of that invited user to owner. 
curl -X PUT https://api.box.com/2.0/collaborations/1234 \
    -H "authorization: Bearer <ACCESS_TOKEN>" \
    -H "content-type: application/json" \
    -d '{
      "role": "owner"
    }'
Depending on the settings, a user with access to a folder might be able to invite other users, yet in all cases only the current owner of a folder can transfer the ownership.