It's not automatic and it's another task all together but during our busy season this year started a cleaning calendar. The calendar is kept on Google Calendar and notifiies my cleaner when modified. In our application, it's a tool not only for when to clean but for other notes like who's coming in next, special requests, remiders, etc. She can view it anytime on her phone or computer.
I suppose you could export the AirBnb calendar to Google but that will show the actual bookings... not necessarily your cleaning dates.
I'm not sure this method is the best, but was pretty handy during heavy activity. ~Susan