Edit in JSFiddle

'use strict';

var Star = React.createClass({
    getInitialState: function() {
        return {x:30, y:30}
    },
    setPosition: function(x, y, refs) {
        this.setState({x:x, y:y})
        if (this.props.followedBy != null) {
            setTimeout(()=>{
                refs[this.props.followedBy].setPosition(x, y, refs)
            }, 100);
        }
    },
    render: function() {
        return (
            <div style={{position:'absolute', left:this.state.x, top:this.state.y, color:'orange'}}>
                ★
            </div>
      );
  }
});

var Screen = React.createClass({
    stars: 3,
    onMouseMove: function(ev) {
        this.refs._1.setPosition(ev.clientX, ev.clientY, this.refs)
    },
    componentDidMount: function() {
        document.addEventListener('mousemove', this.onMouseMove);
    },
    componentWillUnmount: function() {
        document.removeEventListener('mousemove', this.onMouseMove)
    },
    render: function() {
        return <div>
            <Star ref="_1" followedBy="_2"></Star>
            <Star ref="_2" followedBy="_3"></Star>
            <Star ref="_3" followedBy="_4"></Star>
            <Star ref="_4" followedBy="_5"></Star>
            <Star ref="_5" followedBy="_6"></Star>
            <Star ref="_6" followedBy="_7"></Star>
            <Star ref="_7" followedBy="_8"></Star>
            <Star ref="_8"></Star>
        </div>
    }
});

ReactDOM.render(<Screen />, document.getElementById('container'));

<script src="https://facebook.github.io/react/js/jsfiddle-integration-babel.js"></script>

<div id="container">
    <!-- This element's contents will be replaced with your component. -->
</div>