What is it that you NEED for your story, and we'll work backwards from there. Because frankly the problem I am having is the moment of "DUMP".
A long lost relative can be tracked down - end of problem.
A stranger leaves a sort of 'trail'. Call the police, give them a description - it's their problem. Meanwhile the kids are in CPS.
What is the incentive for your family to take on kids they don't know. WHY would they do this?
It is possible for the family to take on two kids, say infant and toddler, and then 'mainstream' them - but not in a community that already KNOWS the family. The community will know these kids are outsiders. And yes, schools will need some paperwork.
Do you need for the family to hide the origin of the children?
EDIT: Okay -this alternate reality scenario is an important element - PLEASE put that stuff up front.
So, you want this family to 'assume' the title on these kids. You stil have to explain to the community where they came from.
Yes, they will need shots and medical records to enter mainstream schools - is that important? You could just homeschool the kids.
Moving to a new location, will 'hide' the parentage. The new community will assume its all one family, or just assume that the new kids are 'cousins' that you are raising.
Much of the legal requiremenst will vary depending on locations.