Run this sample program :
"
import javafx.application.Application;
import static javafx.application.Application.launch;
import javafx.scene.Scene;
import javafx.scene.control.Label;
import javafx.stage.Stage;

public class FullScreenProblem extends Application {

    @Override
    public void start(Stage primaryStage) throws Exception {

        primaryStage.setTitle("Test");
        primaryStage.setScene(new Scene(new Label("test")));
        primaryStage.show();

        Stage stage = new Stage();
        stage.initOwner(primaryStage);
        stage.setScene(new Scene(new Label("test")));
        stage.show();

        primaryStage.setFullScreen(true);
    }

    public static void main(String[] args) {
        launch(args);
    }
}"

Now click on the sub-stage within the fullscreen stage. As soon as you click on it, the fullscreen stage is disappearing.

Now click on another application. And then come back to the JavaFX application, you can now click/move the sub-stage without having the fullscreen stage disappearing.