Okay, I sort of "fixed" this by making all 4 current news items take up the same number of lines (3). It's not perfect as things like emoticons can affect line height as well, but close enough that the jumps should be less noticeable at least. Note: on small devices like mobile phones it may still not work due to word wrapping, I haven't bothered to look. As part of doing this, I need to move the news item with the staff contact information into a post since it's too much to try distilling into 3 lines without getting cramped up there.
It is not the permanent solution we want for obvious reasons. Admins with better skills/time than me can perhaps look and see if the default style can be updated to make the news section fixed in height rather than based on how many lines a news item needs. This approach of course has its disadvantages too, so I'm not honestly sure that's the way we want to go. It would be interesting to see how many people actually pay attention to the news ticker (ie. for its usefulness), and how many people are getting annoyed by the news ticker's cycling causing the main page to jump. There's no point in working hard on this if for the people who are actually annoyed, collapsing the news ticker section is an acceptable solution.