The torso is held erect by some very important “guy rope” muscles; the psoas, and these having an uneven tension in either the side to side or front to back planes will have a remarkable effect on a rider's ability to sit up straight. The psoas connect from the lowest rib and onto each of the lumbar vertebrae. From there they go down through the abdominal cavity and attach above and to the side of the pubic bone. Other extremely important muscles in this area are the iliacus which join onto the lower psoas and run over the inside of the pelvic wing (hip bone) and then becoming the iliopsoas they carry on down to attach to the underside of the greater trochanter of the femur. This is the bony knobble you can feel at the top of the outside of the thigh.
Each time the horse takes a step forward you have to be aware of advancing with him. Many people when asked to move off from a halt into a walk ( or even worse straight into trot) get left behind; which shows in the rider's body as *C* shape as they collapse in the front line. I believe this stems from straying from the original meaning of what was being conveyed in "Follow the horse's movement". Because of this notion of following the rider waits for the horse to move and then has to try and catch up.
You have to think much more of leading the horse with your body. You are initiating the walk, so take the horse with you. This is where the relevance of the psoas, abdominal and lumbar muscles comes in. We all know what the classical alignment of ear / shoulder / hip / heel should look like, but you have to remember this isn't a static pose. To achieve the alignment in motion means a myriad of constant muscular adjustments.
Once the horse is in motion substitute following the motion with your seat for synchronising with it. One of my most commonly used phrases, as I've found it instantly tells the rider what to do is:
"Give the horse your hips". Also, form Erik Herbermann:
“Turn the hips to water. Become the walk. Become the trot. Become the horse.”
Let the horse move you, but no more or less than is necessary. The seat bones are always in contact with the saddle, but the hip joint (not to be confused with the hip bone that you rest your hands on when relaxing) is located higher up than the bottom of the seat bones. This means that this joint isn't loaded when sitting so it can be suspended and free.