How do flying wings respond to crosswinds?
Short answer: The reason for crabbing (skewed flight, as you call it) during an approach in crosswind is to keep the wings level while maintaining sideslip-free flight. Flying wings are not fundamentally different, because they, too, have stability characteristics similar to those of conventional aircraft.

Airplanes want to be flown without sideslip. Flying in a sideslip (where a component of the airspeed comes from one side) incurs lots of complications:


Some of the lift is also pointing sideways, so the airplane drifts sideways.
Weathervane stability will make the aircraft yaw into the wind. This comes mainly from the vertical tail, but is only one of many components to make the aircraft controllable.
The dihedral effect will cause a rolling moment, lifting the windward wing.
Wing sweep will increase this rolling moment and add more yawing moment.
Since the vertical tail sits upward of the line of symmetry in most airplanes, this also adds to the rolling moment.


Most of these effects are desired, because they make the aircraft easier to fly, but during approach they make it harder to line it up with the runway. If the fuselage is held parallel to the runway, the windward wing needs to be lowered, so some of the lift points sideways into the wind, compensating the side force caused by sideslipping. This is achieved by keeping the ailerons deflected to work against the aforementioned effects, and the rudder needs to keep the airplane from turning into the wind. This is no problem as long as the airplane is high enough. On touchdown, however, such an attitude cannot be maintained, or the aircraft will hit the ground with a part of the wing or an engine nacelle first.

Only by pointing the fuselage into the wind can the course of the airplane be aligned with the runway in a sideslip-free way. This is called crabbing, because when seen from the ground the aircraft's speed has a lateral component, a little like a crab wich is also known for its sideways movement. Now the wings are level, so the touchdown will happen with the wheels first. Unfortunately, now the wheels' rotation direction is not aligned with the runway, and to reduce wear, pilots kick the rudder just before touching the ground to re-orient the aircraft parallel to the runway.

The Lockheed C-5 and the B-52 can rotate their gear struts, which enables both aircraft to touch down in crabbed flight. All other types produce more tire smoke and jerk the aircraft sideways if they touch down this way.