iCloud Photo Library

If you store your moments in the photos app it's recommend to enabled iCloud Photo Library. You can enable it in your settings under "Photos" > "iCloud Photos"

After enabling this make sure your photos app syncs and your moments will be safely backed up between your devices. 

Manually 

If you want to manually back up your moments you need to start with un-hiding all your moments. Edit the timeline you want to backup. Disable hiding your content in the photos app 

After modifying your timeline open up the Photos app and find your album

Make a backup of all these videos. If you want to restore these videos add them back to the album you backed them up from.

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