Typical MySQL replication works when slave is listening to changes on master's binlog. Which, essentially, is a log of all operations performed on the dataset. I did some quick googling and apparently it can be accessed quite easily: http://dev.mysql.com/doc/refman/5.7/en/mysqlbinlog.html so I'd assume that this is how they are doing the replication, or something very similar.

As for migration, don't know, you could export the database and listen to binlog but that will lock table for a bit, depending on the database size. But maybe that's acceptable.

Would be curious to hear from folks with more DBA experience :)