Customer moved the domain to the target and enabled Target Rewriting. The status shows as Enabled with Target Rewrite, however external emails are still being delivered without the rewrite applied.
The domain rewrite required some time to take effect because Microsoft transport rules need time to recognize group membership. This delay is expected when the feature is enabled for the first time. Typically, group membership synchronization may take 30–60 minutes after the rewrite is enabled. Once synchronization is complete, there may still be additional replication delays within the tenant, along with extra time for the transport rule to fully apply.
According to this Microsoft community post: Transport Rule and Mail-enabled Security Group: Delayed Evaluation of Group Members | Microsoft Community Hub, the process may take up to 4 hours.
Please note that the replication time for transport rules is dependent on Microsoft.