Apptoto is able to notify booked appointments if the event is moved or deleted from your calendar. When enabled, Apptoto will treat any change in time or calendar of the event as a new appointment; the deletion of the appointment would trigger a notification to the client that the event is no longer active. This option is controlled in the Settings>Advanced & Privacy section of your account as shown below:
If your workflow is to allow the appointment to be booked and then move the appointment to another calendar and do not want the client to be notified, disabling this option would not interfere with that process.