First, a mea culpa. I had a brain barf, got distracted, and wandered off into stirrups and hooves. No excuses (well, except that I've been sick for a week). So, I'm going to finish the series on Medieval horses and then get back to shoeing on Friday.
So, the destrier. Some of you may recall this as the name of Prince Caspian's horse - which, ironically, was almost certainly a courser, not a destrier.
The destrier was the finest, strongest, and most expensive of warhorses, sometimes called the Great Horse. It's often mistakenly thought that modern draft horses are descended from destriers (not true - draft horses were bred up from cobs and garrons). This same misunderstanding leads to the image of the huge warhorse that the knight had to be helped on - while knights probably did need and use assistance when mounting, their horses were not that large. Surviving Medieval horse armor showed that they were between 15 and 16 hands.
Destriers were used by mounted knights and in jousts and tournaments. They were extremely expensive - the majority of mounted warriors rode coursers or rounceys not because they didn't want destriers, but because a good, trained destrier was so expensive. For the price of one trained destrier you could, at times, buy seven less expensive horses. So, only the wealthiest of knights would have them - and often the wealthiest knights were those doing the tournament circuit. It's possible that many of these valuable horses never actually saw real combat in their lives but were used solely in tourneys.
The exact genetic mix of the destrier has been lost - as mounted combat changed, so did the horses bred by the cavalry. The Friesian is often cited as the closest survivor and is popular with modern tournament jousters (as are small draft horses).
A Friesian stallion. Source: B0rder via Wikimedia Commons.
I would argue, however, that the true spiritual descendant of the destrier is the modern warmblood - the specialist, highly bred and extremely expensive animals seen competing at the highest levels in dressage, show jumping, and eventing.
A modern Hanoverian and his rider give a dressage demo.