In order to transfer progress in Fruit Ninja, the process is different depending on your platform. iOS or Android.
Android Devices:
On Android, we use Google Play Games to sync your data online. This means you can transfer it to any phone by simply keeping signed in to the game account on any device you play on. You can confirm if you're signed in on the settings menu of the game.
If you see "sign out" on this screen, it means you are signed in and good to go.
Apple Devices:
For Apple devices, we do not currently have a cloud backup system for data. If you would like to transfer to a newer Apple device, then please contact our team through the "Help & Support" button in the settings menu.