Moving my other comment to a main answer:
Destriers would have been "short and muscular for power, but agile and good sprinters." 15-16 hands high and 1200-1400lbs describes baroque horses (especially Spanish/Iberian horses, who were extremely popular war-mounts), and stock-type/Foundation Quarter Horses (short but heavy, fast, and in modern times are often called "little tanks" or "brick walls"). This link to the National Foundation Quarter Horse Association has a great photo montage of famous Quarter Horse sires.
A scouting party or a messenger could easily cover 30-40 miles in a day, but according to A Collection of Unmitigated Pedantry, regular travel speed of most armies would be much slower, at 10-20 miles per day. Not because the horses can't go farther/faster, but because in medieval times, they have to keep pace with THE INFANTRY AND SUPPLY WAGONS. What use is covering the maximum distance if you and your horse are gasping for breath, and THEN you've left your food, your backup teams, logistics workers, and baggage behind?
![quarter horse working shot 2](https://cdn.statically.io/img/i.sstatic.net/V1qKa.jpg)
Here's a shot of a modern-day working Quarter Horse from an AQHA News article; this horse and rider would look right at home with the Foundation Quarter Horse sires in the first link. Note that while Quarter Horses are certainly thick in the barrel and wide-chested, they are clearly very fit, and a chunky draft horse or a tall and greyhound-like Thoroughbred would have a MUCH harder time getting almost shoulder-to-shoulder with a calf.
Quarter Horses are famously built for a destrier's "sprinting" since they can reach speeds of 50-60 miles per hour (100-120kph for metric) over a quarter-mile's length, so most races take about 30 seconds. But this is exhausting, so a PRACTICAL sprint (where you need enough energy to avoid getting killed after your first five minutes) would "only" be around 30-40mph / 60-70kph.
Despite their heavy weights, working Quarter Horses should NOT look chubby like some modern draft horses, or "bodybuilder-defined" like modern halter-type Quarter horses (who are about the same weight, but often have notorious health issues like spindly legs and poor body structure). Muscle is a lot denser than fat, so a 1000-pound horse may well look "overweight" if they're out of shape, while a muscular one like the horse above may actually look "thin" to people who are only used to seeing pleasure-riding horses that go on trail rides twice a week.
Quarter Horses and the United States' ranching tradition are notably descended from "Spanish" horses and ranching, of which the most famous is the modern Andalusian. "Spanish" horses were extremely popular among medieval nobility for their bravery, their good looks, and agility, and also known for... bull-fighting.
WARNING, THERE IS INEVITABLE VIOLENCE IN THESE PHOTOS.
![el cid francisco goya](https://cdn.statically.io/img/i.sstatic.net/IOclj.jpg)
This photo is from Wikipedia's "Spanish-style bullfighting" page, a copy of Francisco Goya's artwork.
![lusitano bullfighting](https://cdn.statically.io/img/i.sstatic.net/tSmxn.jpg)
The second photo is from Wikipedia's "Lusitano" page. Technically, Lusitano horses are from Portugal, but they and Andalusians are well-known to be closely related.
Note the striking similarities between the bullfighting photos in Europe, and the Quarter Horse roping cows in America. The bullfighters have prettier clothes, but none of the pictured horses are exceptionally tall or draft-like; moreover, they're all sprinting TOWARDS or NEXT TO a charging bull/calf instead of AWAY, so personality and athletic training clearly trumps "immense size." Replace the angry cattle with angry humans, and you've got a skirmish.
Meanwhile, the website helpfully called "Destrier" has a detailed description of warhorses:
"Contrary to popular belief, the medieval knight was not mounted on a huge plodding draft horse like the famous Shire. By combining archaeological finds with pictorial evidence we can paint a picture of the typical Destrier as a very athletic short-backed horse not exceeding 15.2 hands in size.
Medieval paintings often show horses in perfect collection with beautiful gaits, permitting the conclusion that the medieval warhorse was highly trained and extremely manoeuvrable."
Here's a Facebook photo link to one of their albums. In general, "Destrier" shows a whole range of average-height horses that still have the personality AND body-type to get the job done. Some of them do seem like cob-types or smaller draft horses, but the largest modern-type drafts that are 18 hands and twice as wide as normal horses? Notably absent.
Why is that? For medieval safety-precautions, that's why! Most knights preferred short horses because it was easy to mount up, and contrary to pop-culture depictions of knights getting hoisted onto their tall and bulky draft horses, combatants prided themselves in vaulting straight into the saddle, sometimes without touching a stirrup. If they DID end up needing help (such as after getting wounded or exhausted), I imagine they just needed a step-stool, a log, or a boost up from someone else--not something that civilian horse-riders would find unusual.
Here's another Facebook link to a group shot. The note that destriers are usually 15 hands high (five feet tall) and "don't exceed 15.2 hands" would probably not be a hard-and-fast rule, since that extra 2 inches for 16 hands (54 inches) isn't a lot of difference. While five feet and a few extra inches SEEMS pretty short at a human teenager or small adult's height, remember that actual 15-hand horses are five feet tall AND five feet long, plus their necks and heads are famously mobile and often make them SEEM to tower another foot or two above a human's head; in the second shot I linked, most of the riders are adult men in full armor, but do NOT seem "too large" on their mounts.
Also notice how most of the destriers in that shot don't have armor? This is why they don't need to be excessively huge. Jousting armor was notoriously heavy, but it was ultimately for a sport where people did NOT want to get killed (too) easily, and they definitely wouldn't want their expensive HORSES killed if they could help it. A set of human combat armor is often considered to be half the weight of jousting armor, because combatants traded more mobility and speed for less protection/bulk.
Imagine how expensive a full set of horse armor is. And even if you can afford it, who's going to carry it, repair it, and clean it OUTSIDE OF BATTLE? Not your everyday man-at-arms or a noble's second or third son, who may be fine with his loyal horse and his set of weapons/armor, but often needs a job in a richer noble's garrison to MAINTAIN that horse and armor. That's usually how most castles filled their cavalry positions in the first place, as junior nobles and richer commoners were all over the place.
I imagine that real-life soldiers facing cavalry would NOT see a herd of 15- or 16-hand destriers and think, "that's the height of a thirteen-year-old girl, they're not THAT big!"
I can't find the reference at the moment, but it is a notorious problem in period-pieces that when a FAKE cavalry-charge is being filmed, the actors playing foot-soldiers constantly break formation. They know they're not in actual danger, and that the riders won't actually smash into them, but they still can't fight their self-preservation instincts to run away from a wave of galloping horses. Which, if I remember right, is basically the point of cavalry as shock-troops / morale-breakers.
The last photo link is here, and it's a shot of a foot-soldier "fighting" several cavalry members in one of Destrier's reenactment battles. Notice how all these "average/short" horses still basically dwarf him and his shield, and he wouldn't have much more advantage if he stood up? In real life, this guy would be "brave for his last thirty seconds." There's a REASON infantry holding the line against cavalry is rare, after all.