Sliding mesh interface - problem with velocities
 

I have this relative/absolute velocity problem, I think. When in the stationary domain, velocity vectors near the interface point from left to right. While on the other side of interface, in the moving domain, they seem to be opposite. Now, I know, this is wrong, but is that a problem of specifying cell zone conditions or boundary conditions? How can I deal with that?
[see photo attached]

Also connected to this case
 

I tried to star over with building the domain just to check if it's the case of just setting up everything again.
To my surprise, in the simplified "just-to-test" domain the pressure contours in the results are still wrong. They seem to show, that there is no flow between the cell zones of the first and second row and also, that there is a tremendous pressure buildup in the first zone. The vector field of absolute velocities does not differ from the relative one.

Please, help!