Kettlebell carries, whether suitcase style (by your side) or overhead are great exercises to develop spinal stability, as well as scapular-thoracic stability. It's about as simple an exercise as there is...hold a heavy kettlebell by your side (or overhead...I'll get to the technique in a moment) and go for a walk. Focus on staying still and smooth....like you are trying to glide across the floor. Carries are more or less dynamic side bridges, forcing you to resist motion around your spine (the primary function of the "core" muscles) and at the same time gives you great bang for your buck hip control. The overhead version, which I implement as a progression, adds some difficulty to the core stability, while at the same time really challenges scap. stability....which is key to shoulder health.
The keys are as follows:
In the suitcase version, shoulder blades should be pulled down and back. Think BIG posture. Maintain that position while you walk as straight a line as posible. Once you get the basics down, focus on diaphragmatic breathing. We usually walk about 20 yards, switch hands, and walk back.
In the overhead version, the up shoulder should be "packed" while the down scap should be pulled down and back, just like in the suitcase style. "Packed" will mean slightly different things to people in different circles, however the basics are to have your arm extended straight up, palm facing forward. In that position, pull the arm down into the shoulder girdle. If your arm is straight and you have a few inches between your bicep and your ear, you are in pretty good shape. Keep it there the entire time. Last but not least, keep your knuckles pointing up at the ceiling. There should be no break in your wrist, meaning you have to actively hold the Kettlebell in place, not just let it dangle on the back of your forearm.
In the video I walk through a few progressions, starting with simple suitcase carries, followed by the overhead version, then I combine the two for an overhead/suitcase walk (heavier KB on the down side). Following that, I begin to implement lateral steps and a carrioca walk to continue to challenge the scap stablizers in a different plane, and finally I put all the directions together into a "square" pattern, where I walk forward, put the breaks on, carrioca walk, walk back, and carrioca walk to the start. Don't forget to do both sides!
These are just a few versions and progressions my athletes go through to address core and scapular-thoracic stability.
(BTW the video is about 3 min long, and I walk out of the frame a few times...whoops!)