Let's say all this happens inside an accelerating rocket. (Or inside a rocket hovering in a gravity field)Let me say the balls free fall from the same point and same time with various horizontal speed, they hit the ground simultaneously in the train station (pseudo) IFR. In y-moving (pseudo) IFR, i.e. train IFR, the balls hit the ground in different times. By this experiment we can identify which y-moving (pseudo) IFR is "at rest".
1: When a free falling (= inertial) observer looks at a free falling ball, he sees that the ball is moving inertially.
2: When a free falling observer looks at the rocket's walls he sees that the walls are moving non-inertially. The walls accelerate downwards. If the walls have any horizontal speed that speed is affected by the gravitational time dilation that is increasing as the walls are moving to lower gravitational potential.
If we want to know how a falling observer sees the distance between the walls and the falling ball changing, those two rules above give us the answer. Right?
Oh, now I understand that "pseudo-inertial frame" is probably the frame of an observer standing on the surface of a planet. So I talked about a very different situation.