Using labels (or generally GUI components) to represent quickly animating, dynamically created/moved objects is not practical. Its like using a car to drive from the living room to the kitchen.
The Swing components come with a lot of nice features, e.g. they can determine the space they require and a layout manager can use that to automatically place them nicely. Unfortunately for a game, you neither need all the nice stuff, nor do you want to pay the processing overhead involved with it. Matters are further complicated by swings multi-platform capabilities which mandate that each component has a peer component (the peer deals with rendering the component to the active look).
To solve this, you do not represent individual game objects as swing components. Instead you use a single swing component (usually JPanel) as a canvas to paint your game objects on to. You simply call repaint() for that JPanel periodically (e.g. 30 or 60 times a second). Of course you will override paintComponent() on the JPanel to do paint all the game objects using your own code.
Edit: To outline how you do your own rendering
You need to decide how to organize your game related objects. A very simple skeleton could look like this (getters/setters omitted):
public class GameObject {
public int x;
public int y;
public Image image;
}
In practice this class will have additional members (like a behavior that controls how the object moves, and maybe some states/counters for animation). It may also be abstract with a concrete subclass for each possible behavior (like ship, asteroid and bullet). All instances of GameObject are managed in some kind of collection, I assume you use a List.
When implementing the paintComponent it comes down to just loop through the list and render each objects's image:
public void paintComponent(Graphics g) {
List<GameObject> list = ... // however you obtain it in your game
for (GameObject gobject : list) {
g.drawImage(gobject.image, gobject.x, gobject.y, (ImageObserver) null);
}
}
More elaborate games may order the gameobjects to get control of how game objects overlap each other. But in principle its incredibly simple.