“You don’t have permission to take that action on this record or related records. For help, contact your Salesforce admin.” Exception occurs due to the following

1. Check whether you are using Status-Based Routing Model. If yes, check the following Known Issue.
https://trailblazer.salesforce.com/issues_view?id=a1p4V000002JRhrQAG&title=insufficient-access-rights-on-cross-reference-id-when-attempting-to-update-case-records

2. Make sure the affected user have Transfer Records Permission.

3. Setup debug logs to find whether the issue is due to update on different record due to other automation like trigger, flow, etc.