Hi, Hylke.
All the events, organizers, and venues are custom post types so all that should export/import pretty solidly via the WordPress Importer.
PRO’s recurring events possibly wouldn’t transfer over as well because they use the parent-child relationship and I’m not certain those Post IDs would transfer over properly. It’d really be a trial and error thing.
However, if you just push Staging to overwrite Production, that should probably work in all situations.
Of course, we always recommend restorable database and file backups before modifying anything on your site.