Records assigned through Omni-Channel don't trigger any automation like Workflow Rules, Escalation Rules, Triggers and Milestone.

When work items are routed and accepted by Agents through Omni-Channel and if there are Workflow Rule field updates, Approval Process field updates or Entitlement/Milestone suppose to trigger on that object, the automations defined will not fire. However after the record has been assigned and you edit and save the record again, the rules and automations will fire as expected.

Similarly, Escalation Rules will not be executed when a Case is Accepted via Omni-Channel. However after the record has been assigned and the record can be edited and saved again, the Escalation Rules will fire as expected.