First thing to understand here is when something is updated, the index updates itself, not rebuild. That's why when you go to the index in the Smart Search application, you will see 2 time stamps, last update and last rebuild.
Content updates via content staging would trigger index update automatically. If it's not updating, then you should look at the PROD > Smart search > Tasks, are there index tasks stock in there? Is the PROD index updating if you make an update directly? Are you using web farm in PROD?