Do planets on the boarders with stars orbiting just a tad faster than the galactic norm cross the boarder often?
the galaxy is not a single object in space like a planet is, -and some planets have atmospheres which rotate at different speeds than the planets do, and some planets have liquid cores which rotate at different speeds than the solid outer parts of the planets.
Instead, a galaxy is like a solar system, and most like the asteroid belt, but on a much vaster scale and countless many times as many individual objects. Each star in the galaxy is like a separate asteroid, orbiting the center of mass of the galaxy in its own separate orbit.
The idea that stars in the galaxy all obit in step with each other so their relative positions never change is totally flale.
The stars have velocities of tens and hundreds of kilometers per second relative to other stars. It is the only the incredible vastness of space which makes the process of faster stars orvertaking and passsing slower stars as they orbit take many millennia. .
Here:
https://en.wikipedia.org/wiki/List_...own_dwarfs#Distant_future_and_past_encounters
is a link to a list of stars which are cacclulated to have passed with 5 light years of the Sun within the last 3 millin years, with the dates of their closest encoutners and their present distances, and also stars which are predicted to pass within 5 light years of the Sun within the next 3 million years.
This table shows that stars do not keep the same directions and distances relative to each other but are constantly changing their relative positions.
So if the vertical plane separating the Alpha and the Beta quadrants goes through the center of the Sun, it will move as the Sun orbits the center of the galaxy. Stars moving in the same direction as the sun,behing it but faster than the Sun and the quadrant border will eventually pass from one quadrant to the next, while stars travelling ahead of the Sun but slower than the Sun will eventually be passed by the quadrant border.
If teh quadrant borders don't move, then stars will pass from one quadrant to another even more frequently.