Does the Airbus A350 have an Automatic Emergency Descent System?
Airbus will have it on the -1000, and will also introduce it on the -900.

A350 chief engineer Alain de Zotti says the -1000 will be able to &quot;slightly deviate&quot; from the flight plan, if the system is activated, and leave its cruise altitude.
As well as executing the high-speed manoeuvre the aircraft will automatically notify air traffic control of the action.

(flightglobal.com, Nov 2017)

The Falcon 7X already has it, they call it Automatic Descent Mode (ADM).

In case of a depressurization at high altitudes, the Autopilot and Auto throttle will guide the aircraft to safe altitude at maximum velocity. ADM is activated when Autopilot is engaged, the aircraft is above 30,000 ft., and the cabin altitude is 9,700 ft. or greater.


Further reading: Automatic Emergency Descent System (skybrary.aero)