The problem coming in on CX to TBIT if you are flying in a premium class or have elite status is that CX will put priority tags on your bags at your place of origin, which will mean that your bags usually come out first at TBIT.

However, if you are heading onward on AA, the AA baggage handlers don't pay attention to the CX priority tags. You'll need to get the priority AA orange tags so that your bags come out first at your final destination. I've never found the AA bag drop-off area at TBIT to have the orange tags, thus requiring me to head over to the AA terminal with the bags and stand in line to get the check-in agent to affix the orange tags.