On Heroku, I am trying to copy the production database into my staging app using the pgbackups addon. I followed the instructions on the addon page: https://devcenter.heroku.com
UPDATE: This no longer works. Please refer to @lucas-nelson's answer below.
So things are even easier now .. checkout the transfer command as part of pgbackups
heroku pgbackups:transfer HEROKU_POSTGRESQL_PINK sushi-staging::HEROKU_POSTGRESQL_OLIVE -a sushi
https://devcenter.heroku.com/articles/upgrade-heroku-postgres-with-pgbackups#transfering-databases-between-heroku-applications
This has worked beautifully for me taking production code back to my staging site.