This is a non-issue, since you must STOP moving once you cross the border. In the further turns, the rules go back to normal.
That's not correct. The rules also state:
The Anglo-Scottish Border is a heavy red or black broken line. Blocks entering England must stop. Blocks entering Scotland must stop only if they cross the red broken border into Teviot.
So, they only have to stop if they cross the red part of the border.