The initial time they come from would be after 1900. And the different children would come from different years.
These children would have control of their powers and could keep them hidden if needed.
The public doesn't know about these children.
They are being maintained by a few rich members of society.
Their current situation is nice but it doesn't provide the children with a family.
Would these children fare well in a special foster home? Perhaps one that knows about their powers?
The children are currently concentrated in an institution together that helps them be introduced to current society and helps them control their powers.