Why a bird pushes it's wings forward on the down-stroke
We are all familiar with diagrams like this, a cross-section of an aerofoil, they are used to describe how lift is generated. This is also the shape of a bird's wing (image) and it's this mechanism that enables a bird to stay aloft. The physics behind it is complicated and I won't go into that here, but what is important to how birds fly is that the faster air passes the aerofoil, the more lift it creates.
In this diagram I have composited the the up and down wing positions of this barn owl. If we imagine that in the time between these images the body of the owl has traveled forward the distance 'x'. We can see that by pushing it's wings forward, the leading edge of the wing (and the aerofoil) has traveled a distance 'y'. Therefore, the overall speed of the wing edge is x + y. The speed of the air traveling over the wing is actually greater than the speed of the owl's body, thus the owl has very efficiently generated more lift.
The forward movement of the bird comes from the angle of the wing as the following diagram shows.
Also notice that the upstroke, when the wing is partially flexed also creates lift, but arrests forward movement.
As a general rule the slower a bird is flying, the further it'll push it's wings forward in the down-stroke to generate extra lift and the faster a bird is flying the less it'll push it wings forward.
This pelican is a large bird, and as it comes into land, has very little forward momentum to give it lift. It pushes it's wings forward a great deal.*
This cockatiel on the other hand, is small and flies very fast. It doesn't need to generate extra lift to stay aloft and so pushes it's wings forward only a small degree.*