In our org. Omni-channel is set for routing of cases. For Case object trigger is also written. So omni-routing feature assigned a case from the queue to agents, but for that update on ownership of case record, update trigger is not fired.
Found that trigger & workflow rules do not fire for any updation by Omni-Channel. These functionalities are restricted by Salesforce. Because when the omni-channel feature was built salesforce people wanted to make sure that their customers didn't run into any recursive loops from workflow rules or apex triggers firing. Please check the link--(https://help.salesforce.com/apex/HTViewSolution?id=000228601&language=en_US)