React Router v4
There's a couple of things that I needed to get this working smoothly.
The doc page on auth workflow has quite a lot of what is required.
However I had three issues 
- Where does the props.historycome from?
- How do I pass it through to my component which isn't directly inside the Routecomponent
- What if I want other props?
I ended up using:
- option 2 from an answer on 'Programmatically navigate using react router' - i.e. to use <Route render>which gets youprops.historywhich can then be passed down to the children.
- Use the render={routeProps => <MyComponent {...props} {routeProps} />}to combine otherpropsfrom this answer on 'react-router - pass props to handler component'
N.B. With the render method you have to pass through the props from the Route component explicitly. You also want to use render and not component for performance reasons (component forces a reload every time).
const App = (props) => (
    <Route 
        path="/home" 
        render={routeProps => <MyComponent {...props} {...routeProps}>}
    />
)
const MyComponent = (props) => (
    /**
     * @link https://reacttraining.com/react-router/web/example/auth-workflow
     * N.B. I use `props.history` instead of `history`
     */
    <button onClick={() => {
        fakeAuth.signout(() => props.history.push('/foo'))
    }}>Sign out</button>
)
One of the confusing things I found is that in quite a few of the React Router v4 docs they use MyComponent = ({ match }) i.e. Object destructuring, which meant initially I didn't realise that Route passes down three props, match, location and history
I think some of the other answers here are assuming that everything is done via JavaScript classes. 
Here's an example, plus if you don't need to pass any props through you can just use component
class App extends React.Component {
    render () {
        <Route 
            path="/home" 
            component={MyComponent}
        />
    }
}
class MyComponent extends React.Component {
    render () {
        /**
         * @link https://reacttraining.com/react-router/web/example/auth-workflow
         * N.B. I use `props.history` instead of `history`
         */
        <button onClick={() => {
            this.fakeAuth.signout(() => this.props.history.push('/foo'))
        }}>Sign out</button>
    }
}