"Agents are not available. Try again later." Why are these Messaging Session Stuck in Waiting, Won't Route to Agent, Can't Be Reassigned, & Can't be Ended?

In my MIAW setup (synchronous), a customer receives "Agents are not available. Try again later," when a Messaging Session is received during business hours and no agents are available or are busy. 

When an agent becomes available, these Messaging Sessions are not routed to them. Why? 

These Messaging Sessions are owned by Automated Process, stuck in a Status of Waiting, aren't re-routed in Omni-Channel when an agent comes online, have no option to change the Owner from Automated Process, and can't be ended or deleted. 

What are we supposed to do with these records? How can I get them rerouted to an agent, end them, or delete them? 

Please note, this is specific to Messaging Sessions where the customer received the "Agents are not available. Try again later."