Was looking for a way to back up MySQL RDS to local copy to do some work a bit faster offline.
If you want to use replication to move data from one database to another - if you have the appropriate network access to do so:
http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/MySQL.Procedural.Exporting.NonRDSRepl.html
I used another option since my database was pretty small and I run my database access through a bastion host.
I have set up network access to RDS database from the bastion host and installed MySQL Workbench on it. I connected to my RDS instance in MySQL workbench, then click on "Data Export" in left menu and followed the prompts to dump a file.
Then I compressed and copied the dump files to my local machine.
Then reversed the process with local copy of MySQL workbench and choose "Data Import/Restore" on the local machine.
This is a quick hack. If doing this repeatedly would optimize and automate.