When it's somewhat complex or not immediately obvious I have messaged users previously when moving a thread with a quick explanation or if I suspect they might not know how the forum works exactly, e.g someone who might not know how to check their post history to see where it has moved. I don't think every single thread that is moved though requires an explanation sometimes it's something straight forward, and its not that big of a deal.
However, sometimes when moving threads some moderators do include a little note when leaving a redirection thread. Whether moderators personal message a user or leave a redirection thread is case specific, and ultimately comes down to moderator discretion.
However, having said that I wouldn't be against having some sort of automated system, which messages users where it has been moved. Although, I'm not sure if that would become irritating to regular users.
People that spam the forum and get many of their posts deleted by a moderator and get notified for it are not worth the notification. If their posts are deleted more than 50 times, they will be notified that their posts have been deleted. Each posts deleted would be notified. So what makes getting notifications for move and locked topic different? In my opinion, it is better. It is a good idea if we have it.
I think everyone has got a post deleted every so often. Sometimes it just happens because other posts you were engaging with were removed. It's also easy to get off topic every so often especially when an interested topic comes up, but ultimately ends up getting way off topic. Getting the odd post deleted doesn't mean that you're a poor quality user, just means you're human.
At all ramifications, you have made valid points and that seems the moderators is acting friendly with the users having the time to let the OP of the thread in some ways on while the Moving or Locking of the threads action is being carried out.
In my opinion, I other to save moderators that stress, I think the system such apartment such be automated which means the signal should come on the notification box. That is aside from users being DMed or PMed.
Just as made obvious that it would help unknowledgeable users such as beginner users of the forum a lot and that as well would some points aligns the user to under that there are specific sub-boards that is basically for a specific discussion after learning the "Moved" of their thread because we have seen users making threads and appealing they don't know if the board is the right place to post the thread.
Also detailed as action of users spamming or some sorts of irrelevant or outdated threads being "Locked".
Maximizing a deep thought of this would sure seem lot of meaningful benefits for users.