I’d recommend to use mysqldump to generate a SQL file for your existing database, then executing that .sql file against the new DB instance. You can either use exec on the existing container for this, or simply run a new container.
Something like:
$ docker network create dbnet
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$ # Assume original DB was run on dbnet network
$ docker run -d --name db --net dbnet -e MYSQL_ROOT_PASSWORD=password mysql
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$ docker run --net dbnet mysql mysqldump --host db [options] >backup.sql
$ # Copy the .sql file to the new (Azure) VM
$ # Assume our new DB container is also named "db"
$ docker exec db mysql < backup.sql