You definitely have a few options here. The first suggestion that comes to mind is configuring a Rule in Front that performs a Send to Webhook action. You can configure the rule to trigger on any outbound message / outbound reply, and send details of that message to your webhook server, so you can process those messages in real-time as they are sent.

Another option is regularly polling the /events endpoint, filtering specifically for "outbound" or "out_reply" event types to catch all outbound messages.

A third option is to regularly generate analytics exports, which list various details about the messages being sent/received. You can parse out the information you need from the CSV that is generated from the export file.

I hope this is helpful! 

--Jason