Let's look at this from the employer's perspective.
An employee has decided, by their own free will, to move to another location, obviously because they want to live there. That doesn't matter to you as an employer. But the employee is now asking for a higher salary so that they can afford living in this new place.
Why should it be on you to pay the added cost of living, which only the employee personally benefits from? This doesn't make sense. What are you getting out of this?
Nothing. In essence, you are being asked to foot the bill so someone else can go live a nicer life.
Secondly, does this also mean that if receiving a salary increase is justified when moving to a higher COL area; that employees who move to a lower COL area should have their salaries lowered appropriately? What about if they don't move, but the COL drops in their area?
If the salary should not be adjusted to be lower, then there is an easy cheat: everyone moves to the highest COL for a nominal amount of time, locking their salary to the increase, and then moves back to a lower COL area to reap the financial benefits.
Having said all that, this doesn't mean that you can't approach a conversation with your employer; but it should be an open conversation. For example:
Hey, I'm going to move to [new location]. Unfortunately, I have to account for the fact that [new location] has an increased cost of living. I would love to keep working here, but I'm not able to justify doing so at my current wage due to this increase.
Therefore, I would like to ask if you are willing to agree to a salary increase of [X]. I completely understand that the particulars of my living arrangements are not the company's concern, but without this raise I will eventually have to resign for the sake of my financial health.
Don't tell them what they should or shouldn't do for you; simply inform them of what is likely to happen in the near future.
This puts the ball in the employer's camp, without coming across as an entitlement or a demand. Are they willing to keep you on at an increased rate, or would they rather replace you?
If you're a valued employee, this may go over well. If not, well then you can be equally happy that you're able to look for a job with a better suited COL calculation in your new location.
They might not offer you the exact raise you're asking for, but it's up to you to decide whether or not you're happy with what they do offer, if anything.