Review if you have any table with a large number of rows or with blob fields. I suggested to run the following SQL Command: SELECT * FROM information_schema.tables WHERE table_schema NOT IN ('information_schema','mysql','performance_schema'). During the troubleshooting steps we found that during the mysqldump execution the Azure Database for MySQL reached the 80%-90% of the total memory of the instance. And it was needed to scale up the MySQL Instance to a higher database tier to complete the execution of mysqldump process.
Instead to export of the databases at the same time, try to export per database per mysqldump execution if needed.
Reduce to the maximum the limit active transactions in order to avoid many blockings.