How to apply Animator to different game objects (when animations affect children)

Is there a way to drill down from Animator to the game objects that get animated within each state?

In my case I have a keypad of identical buttons. As you see below each number pad has PadCircle and PadText children:

Here is the state machine for the animator applied to Pad_0.

At present, each animation state affects the colour of the two children. (In future it could affect other properties such as scale too.)

I want to be able to apply this animator to each of the other buttons. I figured that I could create a script that gets the children and binds them to the animations during Awake(). Is this possible? Or is there another way to apply an animation state machine to a complex game object like this?