public class CustomDirectRouteProvider : DefaultDirectRouteProvider
{
    protected override IReadOnlyList<IDirectRouteFactory> GetActionRouteFactories(ActionDescriptor actionDescriptor)
    {
        // inherit route attributes decorated on base class controller's actions
        var data= actionDescriptor.GetCustomAttributes(typeof(IDirectRouteFactory), true)
            .Cast<IDirectRouteFactory>()
            .ToList();

        var list = new ReadOnlyCollection<IDirectRouteFactory>(data);

        return list;
    }
}

public class RouteConfig
{
    public static void RegisterRoutes(RouteCollection routes)
    {
        ...
        routes.MapMvcAttributeRoutes(constraintsResolver, new CustomDirectRouteProvider());
        ...
    }
}