Ok, I attack a battlefield with Azir, Sovereign. When he attacks, I put his trigger on the stack with the intention to move my tokens from the other battlefield to this one. My opponent reacts with Star Crossed targeting my Azir. Do the tokens stay where they are?

Yes, the tokens stay where they are.

Here is the step-by-step breakdown of how this interaction resolves based on the rules for [Reaction] timing and the targeting logic of Azir, Sovereign:

The Sequence

1. Attack Declaration: You attack a battlefield with Azir, Sovereign. His "When I attack" ability triggers and is placed on the chain. You choose which token units to move as part of placing this trigger.
2. Reaction: Your opponent plays Star-Crossed in response, targeting Azir, Sovereign. Star-Crossed is placed on the chain above Azir’s trigger.
3. Resolution (LIFO): The chain resolves in Last-In, First-Out order:
   • Star-Crossed resolves first: It returns your Azir, Sovereign to your hand.
   • Azir's trigger attempts to resolve: When Azir’s trigger finally tries to resolve, it checks for "this battlefield" to move the tokens to. Because Azir has been returned to your hand, he is no longer present on any battlefield. As established in RiftJudge FAQ #2127, if Azir is removed (e.g., returned to hand, sent to base, or killed) in response to his trigger, the ability whiffs because "this battlefield" cannot be identified.

Conclusion

Because the ability whiffs, the instruction to move your tokens is never executed. The token units remain at their original location.

This is my interpretation based on Rule 359.3.f (Reference to "here") and the established Azir, Sovereign FAQ (ID 2127). No official FAQ exists for the specific interaction of Star-Crossed returning Azir to hand, but it follows the same logic as removing him via other effects.