Hi John,
If the variations (what you're calling stability) seem to occur throughout the 10 or 15kHz tones, it could be more of a matter of perception (although all machine's performance degrades as you climb in frequency).
Most of the time I use a software scope to do my alignment. I used to wonder about what I was seeing during the upper freq. test tones (perhaps the same as you?).
If I'm not mistaken, with most digital scopes with auto ranging, the ellipse that we see during the 10k tone is 1/10th the size of the ellipse that we see during the 1k tone so effectively it's a 10x zoom to fill the same number of hash marks on the screen. That would lead one to believe that these variations are larger than the actually are.
If you have one of these scopes, try turning off the auto ranging feature and I think you'll see less variation between frequencies (although it's harder to see and use).
The source of these variations are probably a combinations of mechanical and electrical. I think that imperfections of bearings and scrap flutter caused by the fixed elements are a large contributor. But, the higher you move up the "food chain" to something like a Studer 820, the percentage of these mechanical distortions become less and less.
At that level, issues like the balance of rotating parts (unsprung weight) becomes an issue.