I do laugh at people who say that the poor needing housing benefit should not live in London. The same people would soon be squealing if they had nobody to serve their coffee, clean the wards in hospitals etc.
If that happened, what would be the result. With hospitals shutting and no one to serve food in restaurants or to sell ice cream in theatres, or clean the streets etc, London would very quickly become an undesirable place to live. Businesses would relocate as they wouldn't be able to attract staff, so very soon property prices would fall. Then people would start to move back. The free market will sort it out.
In reality, in order to keep staff, they'd have to be paid more, with the end consumer paying more for their meal or whatever, so a natural form of wealth distribution.