Often people carry tension in the lumbar part of the back. This has the effect of pulling the back forward, sometimes accentuating the natural curve. To me, as an Alexander Technique teacher, this sort of back feels as though the person is dividing themselves in two - bottom half and top half.
With horses the same issue can occur, where the dip in their backs can drop downwards and cut off good coordination between the head end and the tail end. Trying to function with this distortion is really inefficient and will inevitably lead to fatigue, pain and even damage.
Alexander Technique training and dressage training both aim to get an integrated muscular system, with freedom in the neck and support from the legs or hind quarters. In order to achieve this, one of the things we need to focus on is getting the back to 'stay back'.
Practising lying semi-supine for 10 minutes will help your back settle and become better integrated. If you do this just before training your horse it might respond better than it would otherwise as it will be able to feel the difference in your use.
The better you are able to integrate your own back, the better your horse will be able to integrate his.