If my understanding is correct, the only way to update the matchData property of a GKTurnBasedMatch mid-turn is [_currentMatch endTurnWithNextParticipant:_currentParticipant
Interestingly, to address your original question, it appears that saveCurrentTurnWithMatchData still sends out push notifications to the other players, even though the current player's turn hasn't ended!
In my opinion, this seems like a GameKit design flaw- it seems inherently misleading to have a badge show up on an opponent's app icon when it isn't yet their turn. What a bummer to open it and see that it's just a false alarm over and over!
If anyone has figured out a way to suppress these push notifications when calling saveCurrentTurnWithMatchData, I would love to hear how you did it.
UPDATE: I started a new thread to address this issue.