Star Alliance carriers are bound by Star Alliance rebooking rules.

They try to protect you inline (same carrier), then Star Alliance, then OAL in that order. The key is that the rebooking has to be in the customer's best interest. If they tell you the next CO flight is not until tomorrow, and that doesn't work for you, push for something better on *A or OAL.

CO has had compliance issues with this policy, so be sure to push them if they resist. I believe the CO profile is GG REACCOM or something to that effect that the agent can look up.