Salesforce Process Automation rules such as assignment rules, auto-response rules, escalation rules, and workflow rules, aren’t triggered when Omni-Channel routes a work item to an agent and the agent accepts the work. 

But, when an agent accepts the work via Omni-Channel widget and then updates/edits and saves the work item record, automation rules are triggered.

Reference Article:

https://help.salesforce.com/s/articleView?id=sf.omnichannel_routing.htm&type=5

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