This is called Lordisos. Not the typical "swayback" from old age, or old broodmares. Here is an article on it.
It is estimated that less than 1% of the horse population is affected by true lordosis, a genetic disease causing spinal deviations and curvature. Studies have shown that lordosis is an inheritable defect, seen most commonly in the Saddlebred, but any breed can be affected.
In affected horses, an incomplete development of the upper thoracic vertebrae is the culprit of lordosis. As a result, overextension of the joints leads to a growth and conformation defect. Horses can have a 5” or greater drop in their spine below the withers, and often the most obviously affected horses developed the condition very early in life.
Affected horses are generally appear “normal” at birth, but the defect develops between 1 year and 18 months of age. Once the process of spinal realignment begins, it progresses quickly, with a massive curvature in the spine able to develop in a relatively short amount of time. In contrast, late-onset lordosis can occur later in life, but is less commonly appreciated, and harder to differentiate between a true spinal problem, and the aforementioned muscle-induced swayback of older horses.
Despite the often startling appearance of a lordosis affected horses, it is remarkable to find that the horses with this defect function almost completely normally, and can leave productive, useful lives. Similar conditions in humans and small animals are usually synonymous with neurological dysfunctions, such as a lack of coordination and paralysis. However in horses, lordosis does not appear to affect any part of a horse’s neurologic or physical well-being.