Here's the deal according to a DL agent at the Partner Award Desk:

If you're on an SQ award in Biz., you will get Coach on the connecting DL flights unless they sell the flight in Business Class. Examples of such flights are BOS - LAX, JFK - LAX, JFK - SFO, JFK - SAN, CVG - PDX and ATL - PDX. Then, you get Business Elite.