Southwest issued this update statement a short while ago:
Statement Regarding Flight 4013 - Updated Mon., Jan. 13, 3 PM CST
The aircraft that landed at (PLK) Taney County airport in Branson, Missouri, departed at approximately 3:00 PM CST, after a thorough inspection. The aircraft is scheduled to resume regular service later today. We continue to support the NTSB in their investigation to uncover the circumstances which led the Pilot in command of flight 4013 from Chicago Midway to land at PLK, six (6) nautical miles from the Branson Airport we serve. The Captain working the flight is a 14 year Southwest Employee and the First Officer is a 12 year Southwest Employee, having a combined tenure of 26 years with the Company. The pilots are currently on paid leave, pending the conclusion of the investigation.
We want to, again, thank responders and Branson Airport Administrators for joining in the work with our ground operations staff to immediately take care of our Customers and their baggage last night. We have since reached out to each Customer directly to apologize, refund their tickets, and provide future travel credit as a gesture of goodwill for the inconvenience.
In previous incidents the pilots have also been put on 'paid leave'. In some cases they were subsequently fired, e.g. the BUR overrun in Y2K and the PIC in the recent LGA hard landing. Often SWAPA is able to get their jobs back as is the custom at a union airline.
And, in case you missed it, video of the takeoff with the usual astute commentary from the news reader:
KSPR special report: Southwest Airlines jet takes off from wrong airport | Local - Home
Looks like they landed at the right airport in Tulsa:
Southwest (WN) #8506 ? FlightAware