BeginInvoke blocks the user interface, but Invoke doesn't.!

I am confused with the scenario that I encountered cross-access. Here is what I am trying to do:

UI main thread - menu item click I create a desktop background and start it asynchronously

private void actionSubMenuItem_Click(object sender, EventArgs e)
{
       ToolStripMenuItem itemSelected = (ToolStripMenuItem)sender;
       ExecuteTheActionSelected(itemSelected.Text);
}

The method is ExecuteTheActionSelectedas follows:

private void ExecuteTheActionSelected(string actionSelected)
{
      BackgroundWorker localBackgroundWorker = new BackgroundWorker();
      localBackgroundWorker.DoWork += new DoWorkEventHandler(localBackgroundWorker_DoWork);
      localBackgroundWorker.RunWorkerAsync(SynchronizationContext.Current);
}

localBackgroundWorker_DoWork It has:

 ActionExecutionHelper actionExecutioner = new ActionExecutionHelper()
 actionExecutioner.Execute();

A method Executein this class that has an invoker method that infact calls an event handler in the user interface thread:

 public void Execute()
 {
      // ---- CODE -----
      new MethodInvoker(ReadStdOut).BeginInvoke(null, null);
 }

 protected virtual void ReadStdOut()
 {
      string str;
      while ((str = executionProcess.StandardOutput.ReadLine()) != null)
      {
          object sender = new object();
          DataReceivedEventArgs e = new DataReceivedEventArgs(str);
          outputDataReceived.Invoke(sender, e); 
          //This delegate invokes UI event handler
      }
 }

The UI event handler is as follows:

private void executionProcess_OutputDataReceived(object sender, DataReceivedEventArgs e)
{
    if (_dwExecuteAction != null)
    {
        _dwExecuteAction.ShowDataInExecutionWindow(e.Text);
    }
}

Now there is a cross flow problem :

public void ShowDataInExecutionWindow(string message)
{
     if (rchtxtExecutionResults.InvokeRequired)
     {
            rchtxtExecutionResults.Invoke(new ShowDataExecutionDelegate(ShowDataInExecutionWindow), message);
     }
     else
     {
            this.rchtxtExecutionResults.AppendText(message + Environment.NewLine);
     }
}

Invoke , BeginInvoke . , , .