I am using SP13 as an incident tracker. As new incidents come in I want to be able to send them to 1 of 5 subsites. Ideally this would be dependent on the administrator filling a column with certain criteria, like the name of the subsite, which then triggers the move, but I am open to other options because frankly there isn't much. Any idea how this can be accomplished?
To me, your solution sounds like a valid plan. Create a SharePoint Designer Workflow that executes on change and looks at the value of that field, then if it has a value it moves copies the item to a incident list in the appropriate subsite. You can then either delete the original item or change it's status in some way to indicate it's been moved.
In the case of moving the item to a list in another site, you can still use workflow to do it, but it is somewhat more complex. You would need to use the "Call HTTP Web Service" action of a SharePoint 2013 Workflow (the action didn't exist in 2010). Then basically you create a REST call to use SharePoint REST endpoints to create an item in a list of your choosing.