Think of it from a particle kinetic theory perspective. Pressure is just particle collisions -- and those impart a momentum. This video helps to visualize what is happening at the micro level (with some simplifications). You can derive all of Navier-Stokes from this fundamental approach.
The summation of forces from all of these individual collisions results in a lift vector that is a function of airfoil shape, relative speed, and angle of attack. Whether you capture this on the macro level in the form of pressure, streamlines, or any other method, the fundamental process is momentum transfer and equal reactions (via Newton - a net mass flow downward for vertical lift). This is not necessarily "downwash" that you are thinking of on the macro scale, as Hank and MikeOH allude to (to be a little more clear here: as the particles collide and transmit their momentum, there is an effective downward motion of a parcel of air on the macro scale).