To reproduce click the "Run" button.

import com.sun.javafx.runtime.VersionInfo;
import javafx.application.Application;
import javafx.scene.Scene;
import javafx.scene.control.ListView;
import javafx.scene.layout.HBox;
import javafx.stage.Stage;
import static javafx.collections.FXCollections.*;
import javafx.collections.ObservableList;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.geometry.Orientation;
import javafx.scene.control.Button;
import javafx.scene.layout.VBox;

public class Main extends Application {

    private static ObservableList<String> data_first
            = observableArrayList("un", "deux" , "trois");
    private static ObservableList<String> data_second
            = observableArrayList("un", "deux" , "trois");

    private ListView<String> lv_first;
    private ListView<String> lv_second;
    
    public static void main(String[] args) {
        launch(args);
    }

    @Override
    public void start(Stage stage) throws Exception {
        stage.setTitle(VersionInfo.getRuntimeVersion());
        stage.setScene(createScene());
        stage.show();
    }

    private Scene createScene() {
        HBox root = new HBox();
        lv_first = new ListView(data_first);
        lv_first.setPrefSize(200, 200);
        lv_first.setMaxSize(200, 200);
        lv_first.setOrientation(Orientation.HORIZONTAL);
        
        lv_second = new ListView(data_second);
        lv_second.setPrefSize(200, 200);
        lv_second.setMaxSize(200, 200);
        lv_second.setOrientation(Orientation.HORIZONTAL);
        
        Button add_remove = new Button("Run");
        add_remove.setOnAction(new EventHandler<ActionEvent>() {

            @Override
            public void handle(ActionEvent event) {
                lv_first.getItems().add(0, null);
                
                lv_second.getItems().remove(0);
                lv_second.getItems().add(0, null);
            }
        });
        
        root.getChildren().addAll(add_remove, new VBox(3, lv_first, lv_second));
        
        return new Scene(root);
    }
}