When a block slides down an incline, why does the incline move back, if the work is done by an external force (gravity)? If I put a point mass on the top of the incline, and if all surfaces are frictionless, I heard, that the incline is going to move back a little (depending on the mass difference), because momentum is conserved and velocity of the center of mass should remain constant. How is this possible though when, it's the gravity (an external force) which is acting on the mass?
 A: The block is pulled vertically downwards by gravity. But it can't move vertically due to the slope. It will have to break through the slope surface in order to do so.
The particles that the sloped object consist of are bonded both vertically and horizontally with chemical forces in molecular or crystalline or similar grids or patterns. Pushing one such particle or atom down vertically, will envoke horizontal forces that pull and prevent the motion in order to hold it in place.
This is where the horizontal forces come from. At the macroscale we see all these forces that together prevent that the object breaks apart merging into what is called the normal force. It always acts perpendicular to the surface, since there will typically be material symmetry across such perpendicular direction. The horizontal component of this normal force represent the horizontally pushing or pulling interchemical forces.
With such normal force pushing on the sliding block, the block will from a similar line of arguments necessarily exert a matching (equal but opposite) total normal force the other way. This is Newton's third law. This normal force also has a horizontal component, and it is this component that pushes the slope sideways.
A: I think you might be considering a special case. If the ramp itself is supported on frictionless rollers at its base, then I can see that what you describe would happen.
However, this is a special case. If the ramp is rigidly fixed to a table which itself cannot move relative to the Earth and is itself rigid then it will not move, except to the extent that it deforms elastically.
A: As @Triatticus rightly mentioned, there is no external force in the horizontal direction. Hence the COM's x coordinate (assuming an x-y plane with x axis along the horizontal direction) should stay constant. Now as the point mass is moving away from, the incline plane has to compensate for that and hence it moves in the opposite direction. Depending on the mass differences, how far both the bodies move can be calculated.

