Quaternion to rotate a GameObject and its children

I have a given Quaternion (with the new direction of one GameObject, calles rotation)
and 3 GameObjects, where each is a child of the other.
I have now by using: GetComponentsInChildren() → transformObjects (length 3).

Now, I want to change the rotation of the parent with my rotation Quaternion, that is given and also all kids in respect to the parent.

In the case, that I say transformObjects[0].rotation = rotation;
It is moving the GameObject far away from the scene and is doing a rotation.

Why, is it moving the GameObject far away, when I use the Quaternion (rotation).
And how can I change the postion of the Children in respect to the changes of the parent?

You can see in the picture above, my 3 GameObjects (a finger). My Quaternion is the rotation of the first Part of the fingern. E.g. it can rotate the finger up and the other 2 parts (chirldren) need to move with it.

Can someone help?

Anything done to the parent affects the children. If you rotate the parent, the children rotate too. If you scale a parent, it will scale the children. Values changed by the parent does not change values inside the children.