You can actually see this here; beehaw recently blocked lemmy.world , so as far as Beehaw is aware, lemmy.world “doesn’t exist”.
As you can see, old posts remain on the instance (unless the admins go and remove them), but new posts don’t get received. I think you might be able to post on Beehaw’s mirror, but they won’t get shared with any other instances.
Of course, this is all subject to change in some future Lemmy version, because this sort of thing can be confusing and counter intuitive.
There is really no way to do that, since these information are stored physically on different servers. So there is no way a disappearance of one server can force another server to delete the content.
This also leads to a disadvantage of the federation, when you delete your account, your instance (probably) will ask other instances to delete your post, even though most instance will do so, but there is no way to enforce the deletion.