When moving an issue from Project A to Project B the moved event is fired for the source project. The parties of the source project's notification scheme registered for the 'moved' event are notified. This notification type is actually a 'moved out' notification type.
It would be useful to also have a 'moved into' notification type. So when an issue is moved a Moved event is fired and JIRA would look up Moved Out notification from the source project and Moved Into notification from the destination project and send e-mails to all relevant parties.