Hat off to Kam Figy for his phenomenally useful post – “Have two web databases? Don’t forget the IndexUpdateStrategy“. The clue is in in the title – if you’ve got multiple publishing targets – so multiple web databases – you’ll need to specify a new (differently named) OnPublishEndAsync strategy (e.g. OnDeliveryPublishEndAsync). The standard one only applies to the Web database. Then reference that strategy.
Go read his post; it’s a good one.