As I understand, you have migrated your app to Azure App Service, now looking to migrate your domains and emails with backups from cPanel to Microsoft 365 or Azure.
For migration, you could follow either of the two options:
Manually export the data at source site or (Export the contents of the source database into an SQL file)
Manually import the data at destination site (upload the new contents of wp-content folder using the File Manager)
Checkout the steps outlined in this doc: Migrate WordPress on App Service on Linux
Additionally,
Map an existing custom DNS name to Azure App Service -This guide shows you how to map an existing custom Domain Name System (DNS) name to App Service. To migrate a live site and its DNS domain name to App Service with no downtime, see Migrate an active DNS name to Azure.
In the doc outlined, you will learn about how to provision the Azure Managed domain in Azure Communication Services to send email: Quickstart: How to add Azure Managed Domains to Email Communication Service
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