do flow builder and omni channel work well together?

I am loving omni-channel, it has really helped us improve our SLA's and prioritisation of tickets. 

However, I have just started using flow builider to improve self service. When a customer needs to speak to support, we are setting different SLA's based on urgency. Some customers are managed 'live' with a 2 minute SLA for first response and other messages are behaving like a ticket with an 8 hour SLA set for first response. We have used triggers to set the different SLA's which is working well. All tickets are being correctly tagged and the SLA set correctly. 

We want omni channel to distribute the messenger and web tickets to our agents based on SLA but this is not working well because all of the tickets that are coming through the messenger channel are presenting themselves to agents as a 'live chat' and waiting for an agent to select the chat, even when its a ticket that we have tagged with an 8 hour SLA. We dont want agents to 'select' those tickets, we want omni channel to assign them at the right time to help us achieve the SLA. 

We want omni channel to identify the live chat tickets and assign them to the person with the most availability and to treat any messenger ticket tagged with a longer SLA to just sit in the queue and wait to be assigned. 

Can anybody help give advice on how you are getting omni channel to work well with flowbuilder? I want to switch all email and contact us forms off eventually and just have messenger/flow builder managing as the first point of contact but i am struggling to see how omni channel works well enough for that.