WebAPI controller inheritance and attribute routing

Check the answer I gave here WebApi2 attribute routing inherited controllers, which references the answer from this post .NET WebAPI Attribute Routing and inheritance.

What you need to do is overwrite the DefaultDirectRouteProvider:

public class WebApiCustomDirectRouteProvider : DefaultDirectRouteProvider {
    protected override IReadOnlyList<IDirectRouteFactory>
        GetActionRouteFactories(HttpActionDescriptor actionDescriptor) {
        // inherit route attributes decorated on base class controller's actions
        return actionDescriptor.GetCustomAttributes<IDirectRouteFactory>(inherit: true);
    }
}

With that done you then need to configure it in your web API configuration:

public static class WebApiConfig {
    public static void Register(HttpConfiguration config) {
        .....
        // Attribute routing (with inheritance).
        config.MapHttpAttributeRoutes(new WebApiCustomDirectRouteProvider());
        ....
    }
}

You will then be able to do what you described like this:

public abstract class VehicleControllerBase : ApiController {
    [Route("move")] // All inheriting classes will now have a `{controller}/move` route 
    public virtual HttpResponseMessage Move() {
        ...
    }
}

[RoutePrefix("car")] // will have a `car/move` route
public class CarController : VehicleControllerBase { 
    public virtual HttpResponseMessage CarSpecificAction() {
        ...
    }
}

[RoutePrefix("bike")] // will have a `bike/move` route
public class BikeController : VehicleControllerBase { 
    public virtual HttpResponseMessage BikeSpecificAction() {
        ...
    }
}

[RoutePrefix("bus")] // will have a `bus/move` route
public class BusController : VehicleControllerBase { 
    public virtual HttpResponseMessage BusSpecificAction() {
        ...
    }
}