B787 pilot reported uncommanded pitch movements and airspeed fluctuations due to severe mountain wave and turbulence.

Narrative

Approaching 70 mi from ZZZ while level in cruise at FL400; the ride was smooth and we received reports of mountain wave and turbulence In the vicinity of the ZZZ [VOR]. Our flight path proceeded southwesterly approximately 30 to 50 mi from ZZZ. About 15 minutes from abeam point; the flight attendants were instructed to take their jump seats and the fasten seat belt sign was turned on. A call was made to the purser to advise them of the situation.Within moments; the ride deteriorated rapidly from smooth with a crosswind at approximately 120 knots to variations in airspeed plus minus 10 knots; followed by a pitchdown and a pitch-up moment. An overspeed warning bell alerted us as the aircraft went into the high speed barber pole; approximately 1 to 2 seconds followed by a level off; at this point I as the PM requested lower urgently from ATC and we were granted FL360. Moments later a second pitch up and pitch down movement the aircraft reacted to the wave and with gains and losses of airspeed of approximately 20 to 30 knots; and the aircraft going into overspeed again as we descended through approximately FL37.5; this time approximately M0.89-M0.92; with a trend going through approximately 325 knots indicated to my best recollection. As we descended through the layer a second request was made for lower from ATC and FL340 was granted. The aircraft was leveled off at FL340 in flight level change; where the ride improved dramatically and no further controllability or ride issues occurred. During the initial and subsequent overspeed; the focus was on ensuring sufficient thrust by my coaching to only use the speed brake to deaccelerate the aircraft and return to a safe speed. I encouraged the autopilot to remain engaged to avoid a Undesired Aircraft State while managing other factors. The descent was requested to ensure the engines had sufficient ability to return to normal N1 cruise values as well as to widen the flight envelope.In addition; maintenance write-ups were submitted through the electronic log book manual code method.