I think "moving forward" is the term you'd more likely see used in this way. It is a sort of expression that means you are acknowledging the mistakes, flaws, or shortcomings of something done previously and that whatever is said after using "moving forward" is like a corrective action to those mistakes that wete recognized. E.g. Moving forward, two people will now be required to patrol the building at all times. Hopefully this will prevent any more incidences of theft. "Going forward" is more of the literal term and is never used by natives interchangeably with "moving forward" E.g. Going forward, we now reach the lobby of the building. As you can see, the design is very compact.