The steps I took to move MySQL from RDS to Aurura Serverless is as follows:
Backup the RDS production instance.
Migrate RDS to Aurora Read Replica instance and promote to master. This is the intermediary instance.
Take a snapshot of the Aurora Read Replica instance.
Restore the snapshot to Aurora Serverless. This is configured to 1 capacity unit at minimum and 32 capacity units at maximum.
Delete the intermediary instance after taking a snapshot (optional).
In this type of scenario since restoring a MySQL 5.6.40 to an Aurora Serverless isn't supported.
An option would be to use the mysqldump utility to back up MySQL 5.6.40 database, then restore it into an existing Aurora Serverless Cluster.
You can use the mysqldump  utility to create a database backup in either SQL or delimited-text format. Once you have created the dump you can load it onto the instance using the mysqlclient or by using a third party tool such as MyLoader.
You should do a test run with each format in a non-production environment to see which method minimises the amount of time that mysqldump runs. As it will depend largely on the size of the database.
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