It's also not that difficult to just do it manually (exporting the production database, and importing it somewhere else), either by mysql command line, or from whatever mysql GUI you might be using. Or just writing a quick PHP script to do it, and then scheduling it as a cron job if you want to make it an on-going thing. Which is probably why you might have a hard time finding something to do this exact specific thing (just not enough of a utility for most people).
Otherwise, Backup & Migrate can be a viable option as others have suggested, for these general types of tasks. If it were me, I would just write up a PHP script to connect to the database, dump the database to a mysql file, then connect to the other database and import. Then schedule it as a cron job. If you search around the interwebs, there's probably code already written for this on stackoverflow, phpclasses or php.net.