There is currently no automatic migration process and not much that is special about "manually" migrating to a C-7 compared to any other hub model, so that's probably why it doesn't exist. 
The long-awaited Hub Protection Service will help with automatic migrations but is not currently available and is reported to cover only Z-Wave and not Zigbee. At the moment, that doesn't really matter since you'll have to do both on your own, anyway.
To move from any hub to another, you'll currently have to do something like this:
- Download the backup from your "old" hub
- Upload (restore) the backup to the "new" hub
- Re-pair and potentially re-associate all devices:
- simply re-pairing a Zigbee device will cause the hub to recognize it as a previously joined device, so this is easy enough; if they were used in any Zigbee groups, I'd open that Groups and Scenes child app and hit "Done" to restore those, however
- Z-Wave is tricky, but the safest option is to pair all devices like new, then swap them out for the old one in all apps, then remove the old device--but see below for a trick that may help.
- most LAN devices should be good (unless they care about the IP or MAC of your hub or something else that may have changed, then you'll need to update those as needed)
For Z-Wave, this is something staff posted a while back that may alleviate some of the above pain, but it's still a manual process. I also don't think it will work if you pair a device with different security to the C-7 than it was with the C-5 (the C-7 supports S2, so that's a notable difference; some devices may also pair with S0 that wouldn't have on the C-5). Parent/child devices also will not work this way. But it may still save a bit of time if you feel like it: