This all depends on where the connection points of the adjustment clips (AC - for short) are.
As far as I can tell, it does not seem to be specific to adjustment clips, but any connected clips should behave the same way (I have not tested this idea thoroughly with other clips, or titles, yet).
Some examples below - clip "P" is in the primary, and clips "A", "B", "C" are connected.
1) Connection point is before the area that will be overwritten:

Pressing Command-Option-DownArrow to overwrite B to the primary, the adjustment clip stays in the same place, as expected:

2) Same disposition, except that the connection point is in the area that will be overwritten by clip B

Before I show you the result, think what you would expect to happen...
Ready?
Do you think that
a) the AC will stay where it is;
b) the AC will move;
c) the AC will disappear
d) the AC will stay but the connection point will change
I have to say that I was expecting c), but instead I got... d)!

So in this case, the effect of the AC does not change.
From several tests, what I get is that the application always tries to keep the position of the AC, adjusting its connection point if necessary, but keeping it connected to the same clip in the primary. Now what happens if the clip in the primary is overwritten completely?
3) In this example, clip "P" is made to completely disappear from the primary, by overwriting A, B and C onto it:

What happens now if we press Command-Option-DownArrow with all three clips selected?
You'd be right if you guessed that the AC will disappear.
Finally, what situations might cause the AC to move?
This happens when the clip in the primary partially remains, but only in areas completely outside the range where the AC is, like in this case:
