As far as I can see Min, there is no horse problem in the Southwest of France. There are lots of healthy looking horses kept around here on large fields and lots of horse meat in the supermarkets. Unfortunately, although horse meat is very tasty, it is very expensive here.
Although I see lots of horses in fields I rarely see anyone riding them around the lanes. In fact I've only seen one person on horseback since we got to Gascony, so perhaps these horses are being kept mainly for meat?