My child's mother lives in Deleware for the last five years but it appears that she is still collecting Ma. Benefits from where she moved. Is this legal.

Child support is not a "state benefit" it is paid by the other custodial parent.
The child support will be paid often though the court where the court order is issed, And if it was issued though that state, it remains there. The court with jurisdiction over child custody and support does not change just because they move.

So if this is support, yes this is how it works