Variable rents, based on income is one way of dealing with this. When you sign the tennancy for a council or housing association home you agree that if your income goes over a certain level that you pay a higher, market related rent.

I think it is unfair to force people to move if they don't want to (although many are, for other reasons) and this avoids the very real problems of sink estates.

The additional rental income can be used to build more council/affordable housing, thus providing new homes for others that need them.

Difficult to police and enforce but non unrealistically so.