Start the iDevice Manager and press in menu "Tones".
Connect the old iPhone with an USB cable on your computer and confirm the "Trust computer" message on the iPhone, if displayed.
The existing tones will be displayed.
Transfer the tones to your computer. See details above.
Unplug your old iPhone from PC.
Restart the iDevice Manager and connect your new iPhone.
Navigate again to the Tones.
Move by drag 'n drop the transferred tones over the (empty) ringtone list and release the mouse key.
Alternatively, can you use the "Add" button in the top menu.