Originally Posted by MSPeconomist

AF has a few flights to Pacific Islands, IIRC from LAX. (DL has a daily 777 nonstop SYD-LAX.) I don't think any Sky Team carrier serves NZ, so you need either to find partners of the airline whose miles you're using (for DL miles, Virgin Australia, which also has SYD/BNE/MEL non stops to LAX; for other SkyTeam airlines, check if there's a mideast partner such as Emirates or Etihad) or to purchase a transTasmanian flight, most conveniently to SYD. (DL miles can also be redeemed on Air Tahiti Nui.)

The bottom line in my confusing attempt at an answer is that SkyTeam miles can be used on any SkyTeam member's flights if available in the right (lowest) award inventory class, but SkyTeam airlines generally have additional partners for redemptions. Some FF programs impose restrictions on whether alliance and other partners can be combined in a single award ticket (DL doesn't currently), so you need to check the rules carefully. Similarly, you need to check what happens if you want to use a codeshare operated by a non partner or non alliance member. RTW award tickets have additional rules, but sometimes it's better to patch together award tickets and even purchase a few segments rather than getting a RTW. Remember that a partner of a partner isn't necessarily a partner, either for earning or spending miles.