And, they do! Long range artillery fire (as from battleships) must take this Coriolis effect into account.
The Brits failed to adjust for changes moving from northern to southern hemisphere in the Faulkland Island war resulting in some very inaccurate gunnery at the beginning.
Another example would be the Paris gun used by the Germans in WWI (from a range of ~75 miles).
It is the rotation of the earth that causes the phenomon. The projectile is traveling in a straight line while the earth turns beneath it.
A train would be subject to the same effects, since a "straight" line really is following the curve of the earth. The ball's trajectory could be computed via Newtonian physics using a vector matrix. It is a combination of the momentum imparted by the earth's spin (think cetrifugal force) throwing the ball in a straight line away from the earth, the forward momentum of the ball (equal to the train's speed and direction at the moment it is released) and gravity pulling the ball straight towards the center of the earth (at the moment it is released).
The ball would not hit the target, although it is simplistic to say that it would land a bit behind it.