Poppy and Snape are worried that with the wards gone that Dudley's a target.  Petunia and Vernon can rot, but they feel bad about Dudley. Poppy's made a Muggle child protection case file for Harry, showing that he's been taken into care because of reports by his school--a Scottish School for the Gifted.

Poppy has been looking for someone with contacts to take the file to London.  Dr. Granger has a friend who's a social worker in child protective services.  She's taking Harry's file (which is quite enough to get Dudley taken into care, all by itself) to her friend and saying,  "I shouldn't be doing this, but I don't want to wait until those people in the Scotland office get off their asses and actually call you about the cousin."  The story being that Scottish Child protection is handling Harry's case.  

The Somebody Else's Problem Charm makes it so they can use the information from Harry's file for Dudley, but since Harry's not their case, they won't actually look for Harry (if they even take note of the boy's name it would be a surprise).  Eventually, the file will get passed around and make it's way to the Muggle liason office addressed to Arthur Weasley.  Somebody Else's Problem charms work equally well on all government workers, Wizard or Muggle.

Foster care is a decent protection for Dudley because Pureblood wizards (and those masquerading as such), won't understand the government putting children with seemingly random strangers.  Even Voldemort will be looking for old fashioned orphanages.