There are couple of different options depending on how things are deployed really, I would start with the Backing up and Restoring section of the manual though as it has some information on this.
If Postgres is already external option 2 feels like it would be the best option.
The first that comes to my mind is to use management commands export_workspace_applications and import_workspace_applications. That would need to be run for each workspace you want to transfer over (this would however not transfer user accounts I think).
However if you can make a whole dump of the database and import that it is probably the best option.