The process is the same. To bring them into supervised status you'll need to wipe and enroll them again. And you can't use any previous iCloud backups in the process. I have had success using the proximity restores when going from device to device.
In these cases I normally recommend you set a date going forward where all devices enrolled will be done through ABM setup. Trying to bring older devices into the fray is often chaotic and not worth the trouble. The only time it is really doable is if wiping the device is of no consequence to the use case at hand. Like Kiosks, and single use devices that have no personal data attached.
I would echo Luke here. Below are some scenarios that may assist getting the users on board though to help streamline the process for you.
-Have a handful of spare devices available as spares and perform a swap program. Bring in your old device and we'll give you a new one. Once the user returns to old device, add that into DEP and into the spare device pool.
-Device refresh. Depending on when you're are planning to next refresh devices, you could do this task on all the newly procured devices.
-Carrot for users. Provide an incentive for them to do it, maybe they get access to new apps and services.