This code can be improved upon and be a basis for an online desktop publishing application, postcard designer, greeting card designer, etc.

<?xml version="1.0" encoding="utf-8"?>
<mx:Application xmlns:mx="" layout="absolute">
 <mx:Script>
  <![CDATA[
   import mx.core.UITextField;
   import mx.containers.Canvas;
   import flash.events.MouseEvent;
   import flash.text.TextFieldAutoSize;
  
   // this will hold the currently selected UITextField
   private var activeUITextField:UITextField = null;

   private function mouseDownHandler(event:MouseEvent):void {
    event.currentTarget.startDrag();
    
    // we use getChildByName() to access the UITextField inside the canvas
    var tempUITextField:UITextField = event.currentTarget.getChildByName("tf");
    
    // remove border of the active UITextField, show border of new active UITextField
    activeUITextField.border = false;
    tempUITextField.border = true;
    activeUITextField = tempUITextField;
   }
   
   private function mouseUpHandler(event:MouseEvent):void {
    event.currentTarget.stopDrag();
   }
   
   private function loadUITextField():void {
    
    // create a new UITextField, set appropriate values
    var newUITextField:UITextField = new UITextField();
    
    // autoSize is UITextField's main advantage over Text and TextArea
    // although you can't edit a UITextField
    newUITextField.autoSize = TextFieldAutoSize.LEFT;
    newUITextField.multiline = true;
    newUITextField.text = "New UITextField";
    
    // set the UITextField's name to tf, will be used later for child access
    newUITextField.name = "tf";

    // create the Canvas container that will hold the UITextField,
    // this will be the one that will handle the dragging
    var UITextFieldHolder:Canvas = new Canvas();
    UITextFieldHolder.addEventListener(MouseEvent.MOUSE_DOWN, mouseDownHandler);
    UITextFieldHolder.addEventListener(MouseEvent.MOUSE_UP, mouseUpHandler);

    // we must add the Canvas to the main application FIRST before adding
    // the UITextField to the Canvas, doing otherwise will cause errors
    addChild(UITextFieldHolder);
    UITextFieldHolder.addChild(newUITextField);

    // set current UITextField as the active one
    if (activeUITextField != null) activeUITextField.border = false;
    newUITextField.border = true;
    activeUITextField = newUITextField;
   }
  ]]>
 </mx:Script>
 <mx:Button label="Load UITextField" horizontalCenter="0" verticalCenter="0" id="btnLoad" click="loadUITextField()"/>
</mx:Application>