Moving Innodb database from Server A to server b
Tables can be moved in physical form, although it is not instant: https://dev.mysql.com/doc/refman/5.7/en/innodb-table-import.html

A more automated way would to use partial or full backups with tools such as mysql enterprise backup or Percona XtraBackup: https://www.percona.com/software/mysql-database/percona-xtrabackup which will allow you to backup a database quickly without impacting sourde database availability.

In the future, having a replica of your database will be the easiest way to increase your availability during maintenance (switchovers should only take seconds).