Private property is so yesterday
If it's private property rights you want, get rid of zoning entirely. If a developer buys the house next door, tears it down and puts in a duplex, or maybe a ten unit apartment building … well, he's the owner, and he has a right to do it.
The suburbs have brought a lot of this down upon themselves with exclusionary zoning and a failure by so many suburban jurisdictions to accept and plan rationally for their fair share of low income housing.
The particulars will vary considerably from one city to another. I live in DC. In this area, the suburbs have operated for decades on the assumption that the District would be the dumping ground for the entire region. Now gentrification is pricing a lot of the poor out of the District. They have to go somewhere. Looking at the metro region as a whole, DC still has more than its share of the poor, but the balance is shifting.
Most freepers would advocate controlling our borders and stopping illegal immigration. That would be a start. But we're not doing that. So here we are. We're pricing the poor out of DC. I hear Annandale and Darnestown are nice. There are a lot of suburban ramblers on nice cul de sacs that could easily house three families. There's even room for multiple cars on concrete blocks in the front yard. Better your neighborhood than mine. I've already paid my dues.
If that doesn't strike you as a good solution, you need to engage constructively in regional planning to spread the burden of low income housing. The poor WANT to move to the suburbs. That's where most of the jobs are, especially the service type jobs for which they might qualify. Crime is lower and the public schools are better. Housing costs are lower. Let 'em move. Let 'em double and triple up, illegally if necessary, in suburban housing. You can't assume they're just DC's problem any more. We're pricing them out. It's the free market at work. It's ok with me if you're the only English speaking person in your carpool.