It depends very much on the direction, shape and speed the object moving. In the case of a plane even in takeoffs and landing the movement is mostly horizontal. With high speeds of the object the air flows from the front of the moving object to the rear. Like in the picture below some air is trapped inside the turbulent eddies behind the object if the air speed is large enough for the flow to be turbulent and not laminar (so it is replaced by turbulent mixing relatively slowly).
To further go to this case of a plane wing this picture shows quite shows what it looks like:
Wind plane lab http://cdn4.explainthatstuff.com/airfoil-wind-tunnel.jpg
Of course this is a laboratory case and the wing is actually in a wind tunnel (wind blows around and wing is still). In real life the air flow is much more turbulent in high air speeds that in this picture, the turbulence is reduced from these kind of laboratory tests. Although this kind of test then represents reality, where the wing is moving and the air is more still, better.
These all of course hold true to other fluids (for turbulence is a attribute of flow itself and not the fluid) with different densities and viscosities when wind speed is also adjusted.