You need to pass a callback to the child component. I just forked your codepen and added some snippet as below.
http://codepen.io/andretw/pen/xRENee
Here is the main concept,
Make a callback function in "parent" component and pass it to the "child" component
i.e. The child component needs an extra prop to get the callback:
<Form>
  <FieldGroup>
     <Field rules={'required'} cb={yourCallbackFunc}/>
  </FieldGroup>
</Form>
In <FieldGroup /> (parent):
class FieldGroup extends React.Component{
  constructor(props){
    super(props);
    this.state = {
      color: 'blue'
    }
  }
  cb (msg) {
    console.log('doing things here', msg)
  }
  render() { 
    const childrenWithProps = React.Children.map(this.props.children,
     child => React.cloneElement(child, {
       cb: this.cb
     })
    )
    return (
      <div class='fields-group'>
        <label> field </label>
        { childrenWithProps }
      </div>
    );
  }
};
In <Field /> (child):
class Field extends React.Component{
  constructor(props){
    super(props);
    this.state = {
      empty: true
    }
    this.validate = this.validate.bind(this);
  }
  validate(e){
    let val = e.target.value;
    console.log(!val);
    this.setState({empty: !val});
    //here to notify parent to add a color style!
    // do call back here or you may no need to return.
    this.props.cb(val)
    return !val;
  }
  render() {
    return (
      <div>
        <input type='text' onBlur ={(event) => this.validate(event)}/>
        {this.state.empty && 'empty'}
      </div>
    );
  }
};
And you can do the things you want in the callback function. (You can also pass a callback from <Form /> to the grandson and get it work, but you need to rethink the design of it is good or not.)