Using our the client tools in your local environment. But, please, remember about the limits of OutBound data transfer because you could have more charges in your invoice.
This time I would like to share with you a lesson learned about how to export a database of MySQL using the Azure File Share creating a folder where saved my exported file and making accessible from anywhere using my credentials:
Phase 1: I created a virtual machine in Azure in the same region that your Azure Database for MySQL is running on.
Phase 2: Depending on the size of your database, you could attach a storage or create a File Share in any storage account. In this case, I chosen to use Azure File Share.