i.e. is it only downward adjustments for COL that you object to? and, if so, what's the logic in that (other than pure self-interest)?
Likewise, if I take a job and get X salary for that job, then move to a cheaper part of town I wouldn't expect to take a pay cut.
You're employed with a salary of $X at location Y.
Your employer wants you to move to location Z with higher cost-of-living. You can either stay at location Y or move to location Z. You feel it's right that your employer increase your salary because of the higher cost of living if you move to Z.
The company is asking you to make a concession: do the same work, but in a different location.
When you ask to move to a lower-cost area, aren't you asking for the same concession? Why is that concession expected to be gratuitous vs the same concession from the company which is expected to be compensated?
On the other hand, if I want to do something because it is convenient for me but does not affect my work in any way (i.e. change where I work from) I would not expect my compensation to change.
If an remote-work employee wants to move to a lower cost-of-living area it should make no difference to the employer. They're still getting the same work as before, at the same cost. If the work output was worth that much before then it's still worth that much when performed from a different location.
If an employee chooses to move to a more expensive location, on their own initiative, then that is more like the second scenario than the first, and there is no reason to expect the employer to offer higher compensation to offset the cost-of-living in the new location when there is no particular benefit to the company from the move.
EDIT: fix typo
New hires generally have their existing compensation as part of their negotiation. So if I'm already living in SF and am negotiating with a new employer for salary, my current high CoL salary gives me leverage to expect higher pay regardless if I'm working remotely. Remote work allows for flexibility but it doesn't erase CoL concerns.