I've received a few op-ups on CX, including one yesterday, never on any other of the OW carriers. It's been handled in a variety of ways, once at ticketing in AMS on a AMS-HKG flight, once a CX rep was waiting for me with a new boarding pass as I got off a flight in HKG and was connecting to SFO, and a few times by asking or responding to an announcement at The Pier/Wing in HKG. I'm AA EXP (OW Emerald), but not part of Asia Miles.

Your chances are the best if you can talk with a ticketing agent, or better lounge personnel, about 1-2 hours before the flight. Depending on where you're flying to/from, this could mean getting ticketed to the first CX gateway only, collecting luggage in transit, and then going to CX ticketing for ticketing for the rest of the journey.