You can use a timer. Put a button on your form and trigger the button_click event for it. Create a timer and trigger the timer_tick event. In the button_click event start the timer. In the timer_tick event use compound assignment operator += to assign and add the text from one text box to the other.
To add a new line each time on top of the text, just add + Environment.NewLine to the end of the statement. But unless you're using a richtextbox make sure you set the Multiline property of your textbox to true.