Any patterns that investigators look for? For example, what happens when the engine flames out? is it guaranteed to enter a high speed dive? And would this affect certain parts of the aircraft, causing them to separate under torsion and not compression?
If a "ghost" airplane with passed out pilots is unattended and the engines flame out, what is likely to happen is it will fly along on autopilot until one engine flames out a few minutes before the other. This will impose a pretty large amount of yaw, which the airplane's yaw damper may or may not have enough authority to counteract. Yaw dampers generally can move the rudder somewhere around 1/3rd of normal travel and aren't meant to cope with asymmetric thrust (I recall from recurrent training in the CRJ that a flameout in cruise still required more rudder than could be applied by the YD to keep it flying straight and you still had to get your feet on the pedals post haste).
With the one engine dead, the yaw damper will be at full travel trying to "center the brick" (the little box shaped skid indicator on the Primary Flight Display), and if that's not enough, the airplane will be skidding along trying to roll into the dead engine, and the autopilot will be applying aileron to try to keep the wings level, and the airplane will fly along cockeyed, like happens when a new pilot on the type course experiences his/her first engine cut and forgets about their feet.
The amount of aileron may cause the autopilot to kick off at some point, at which point the airplane will immediately roll off toward the dead engine into a spiral dive (they won't just dive straight down, they have to roll off to one side into a spiral where the airplane's natural speed seeking pitching tendency just causes it to wind up the turn tighter and tighter as it goes steeper and steeper and faster and faster).
Or, if the autopilot is able to hang on, eventually the other engine flames out, the ram air turbine drops, most electrical services are terminated except emergency ones and the autopilot is likely to kick off then.
At that point, if the airplane is trimmed nicely and the air is smooth, it will likely simply descend more or less straight at its existing trim speed in a straight ahead glide... until something disturbs it, like a bump that makes it bank a bit more than dihedral effect can accommodate.
At that point most airplanes will drift off slowly into a descending spiral. It will turn steeper and steeper and descend faster and faster, either until bits start to come off the airplane from overspeed/over G, or it hits the ground (bits raining from the sky following spiral dives is what usually happens in GA disorientation events in instrument conditions - think Kennedy's crash).
So in the end, one of two things will happen. If the airplane was trimmed at 240kt indicated, it will fly into the ground straight ahead descending at 240kt, or, if it got bumped into a spiral, it will come nearly straight down at close to Mach, making a small crater, or come down in individual pieces after breaking up on the way down.
And ending like the Lady Be Good is unlikely with a jet.