On mind-numbing inconsistencies in momentum transfer, and Galilean invariance

1. Imagine a layer of air moving across a water surface. If we observe the situation sufficiently "downstream", the water surface is moving too. It follows that the velocity of the air and the water at the interface is the same. Air molecules move into the interface with a certain momentum, but emerge, several collisions later, with the velocity of the water at the surface- its the same with the water- so both move at the same velocity.
Now, let's look at the momentum(s). The velocities are the same, but the momentum of the air and the water at the interface are not- Water convects about a thousand times more momentum than the air. (This is the ratio of their densities.) And yet, the net momentum transfer through diffusion is from the air to the water- the air set the water in motion, remember?

2. A fluid flows over a flat plate- and you get to see a boundary layer form. A nice velocity profile develops, and the net result is the transfer of momentum from the water to the plate. This vague expression manifests itself in the gradient of the velocity component perpendicular to the plate. In particular, towards the edge of the boundary layer, the gradient of the perpendicular component of the velocity is greater than zero. (Think continuity, and the fact that the parallel component of the velocity at the edge of the boundary layer decreases along the length of the plate as the boundary layer grows.)
Now drag the plate instead. At the edge of the boundary layer, the parallel component increases along the length as the boundary layer grows. The result, therefore, is that the gradient of the perpendicular component (along the perpendicular direction) decreases.
The reason for the emphasis? The aforementioned situations are, at the face of it, the same, save for a change in frame of reference.
And yet, the gradient of the perpendicular component, which is unaffected by the coordinate transformation, is simultaneously positive and negative!

3. The Navier Stokes Equations are not Galilean invariant. Beats me, but its true. There's a hint of broken symmetry here- and a flaw that is as elusive as it is enigmatic.

I'm being peppered with paradoxes all day long- I'm not surprised I can't get any real work done nowadays.

EDIT: (1) is now resolved. There is no inconsistency. Both thumbs up to Mr. Sambashivam and his penetrating insight into the issue. On the off chance that the garbled, obfuscating, voluble description of the apparent inconsistency made sense to you, and that you're curious to know what the resolution is, comment, and ye shall find the elucidation you seek.