How is a transonic aircraft longitudinal stability different than a subsonic and supersonic aircraft?
At this level, there is nothing special about transonic stability.
Generally, airplanes become significantly more stable at supersonic speeds (due to aerodynamic center moving backward). At the same time, elevator efficiency drops. Transonic is simply the transition between the less stable subsonic regime and more stable supersonic regime.
Now, for an engineer as well as a pilot, transition can be a challenge, especially if it's quick. For example, most supersonic aircraft have variable stick-to-elevator ratio, and transition happens at transonic speeds. Apart from making sure that the aircraft is controllable throughout the transonic range, you need to consider what happens if the system fails at any stage.
As another example, modern fighters are often made longitudinally unstable. But this is only true at subsonic speed. They are all stable at supersonic regimes. Now you have transition between even more different states: unstable to stable (and back). Flight control systems must cope with that.
Thankfully, the transition is not too abrupt, especially for swept wings. (This, controllability, is in fact one of the main reasons to use swept wings. It's not just about drag).