Does the vertical distance change at a constant rate? How do you know?

This question is incomplete. You need more context to learn about the motion.

I will show you some cases for which the vertical distance does not change at constant rate:

1) If object A is moving on a path that is parallel to an arbitrary line, the distance between the object A and the line will not change.

2) If object A is moving at the same rate than it is doind object B, the distance between object A and object B will not change.

Without more information from you this is what I can do and I think it is useful.