What is the best practice for redirecting the user to a new action in the backend? I don't see a controller method for doing this in the API.

I have the same question, exactly 1 year later.

Not sure exactly what is being asked here, but you can return a Redirect from your backend controllers.

use Redirect;
use Backend\Classes\Controller;

class Whatever extends Controller
{
    public function hello()
    {
        return Redirect::to("foo/bar")
    }
}