The latissimus dorsi is a non-weight bearing back muscle.
Origin: The supraspinous ligament from the 3rd thoracic vertebra via the thoracolumbar fascia.
Insertion: into the teres major tuberosity of the humerus.
Function: The muscle is responsible for shoulder flexion and retraction of the forelimb.
Soreness: Because the horse, naturally, does not need to support weight over their back, this muscle and other back muscles are typically weak and require a lot of strengthening to support the weight of the saddle and rider. Tension can build up relatively quickly as a result of regular training, rider weight and balance, ill-fitting saddles and those are 'cold backed'.
Long and low, core stabilisation and pole work exercises can help to engage and strengthen this muscle. Forelimb extension stretches can aid in relaxing this muscle and improve the compliance of the muscle. Pole work exercises that I would suggest for this muscle would be raised outside poles on a curve and it is particularly useful for horses who are weaker on one side.