Gravitational forces from the earth affect all masses in the same manner. In other words, the acceleration due to gravity (g=9.81 m/s*s) is constant regardless of the mass of the object. Thus, if two objects are placed a given horizontal distance apart, that distance will remain constant as they accelerate towards the earth over a short distance. This is because they both accelerate at the same rate and their displacement towards the earth is identical at all points in time.
If one were to assume that the projectile gun would begin falling at the same time as the target, their position relative to each other would remain constant according to this theory. Furthermore, if one of the objects had an initial velocity towrds the other, it would move towards the target at that constant velocity for the duration of the fall. If that were so, a projectile launched from a gun at the same time as the target is dropped, would automatically strike the target. This assumes that gravity begins accelerating the projectile at the exact same time as it begins accelerating the target.