Amazon doesn't have any way of backing up their AWS Cognito User Pools.
cognito-backup is a CLI for backing up the data, and can also restore afterwards. Note: AWS has no way of extracting the passwords of your users so you need to store these separately
Requires node 8 or newer
npm install -g cognito-backup
cognito-backup backup-users <user-pool-id> <options> Backup all users in a single user pool cognito-backup backup-all-users <options> Backup all users in all user pools for this account
cognito-backup backup-users eu-west-1_1_12345 cognito-backup backup-users eu-west-1_1_12345 --region eu-west-1 --file mypool.json cognito-backup backup-all-users eu-west-1_1_12345 --region eu-west-1 --dir output cognito-backup restore-users eu-west-1_12345 Abcd.1234 --file eu-west-1_12345.json