Q: So what exactly happens after the rights and obligations are "transferred" from one set of parents to another?

The natural parents cannot:-

1. legally claim back the child from the adoptive parents - unless they had been cheated into giving their consent - this right to keep the child is called "custody";
2. have any rights, obligations, duties or liability concerning the child - they cannot dictate how the child shall be clothed, fed, housed, and educated - this duty to clothes, feed and house the child is called "maintenance".

By contrast, the adoptive parents must/may:

1. make sure that they see to the child's custody and maintenance (including education)
2. treat the child as their legitimate child - this means that the child is entitled to inherit the belongings of the adoptive parents with equal share as any natural born child/children of the adoptive parent
3. be regarded as the lawful parents of the adopted child if they are legally married;
4. be paid any compensation resulting from the death of the adopted child.

As a result, the child will

1. lose all rights to inherit the property of the natural parents
2. inherit the adoptive parents’ property if they die without a will
3. not be able to marry the children of the adoptive parents as they fall within the prohibited degree of relationship.