Hmm.. This is interesting.. I never realized that you should take the Dragoon that was used at the time when there still existed armoured knights.. And yes, the earliest Dragoons (don't know if the earliest Dragoonlike units were called Dragoons, but the thing that eventually became Dragoon. Lets just call it Dragoon) would have rather poor firearms.. But as soon as 1 handed firearms came, the Dragoons used them. As far as I know a rich or well valued Dragoon could have more then 1 firearm. He could have several one handed guns so that he would be able to fire more then 1 bullet at the knight.
And.. This discussion is of course the unlike event that a single armoured knight would duel a single Dragoon. This could of course happen in the midst of the battle, but if it happened in the midst of the battle the Dragoon would be likely to have spent his bullets already and would then face a much stronger melee unit, so in the midst of the battle the Dragoon would probably loose.
But, say there was a duel. They both start 1000 meters or so away from eachother and are to fight to the death. Lets say its the latest and greatest knight VS a rich and valued and well trained Dragoon. And lets presume that the bullets of the Dragoons guns could penetrate the knights armour at around.. oh.. lets say 50 meters, that's pretty fair I think. And lets say the Dragoon had 3 or 4 guns.
They would move closer and closer, the Dragoon would either have to wait for the knight to get closer to be able to penetrate the armour, or ride away from the knight at a slower speed then the knight so that the knight would get close enough for penetration but still be too far away to harm the Dragoon. If he picks the first solution he would get
good aim, but little time to fire and try to get out of the knights way, and if he picks the second solution he would be rather safe, but have a much poorer aim.
So yes, I agree, the Knight would probably win.. But, if you took the late Dragoon vs a prime knight I think the Dragoon would be better off, with much better firearms, more accurate and powerfull.
As for the equipment: Yes, a horse, a breastplate, a helmet, maybe bracers, 1 or more sword or other melee weapon, a small shield, like a buckler (I guess he could have a larger shield too) and 1 or more firearm.
And yes, the Dragoon was meant to be an infantry who could ride a horse if necessary, so he'd start on foot with a musketlike firearm, and then move on to use the horse and smaller firearms if required. However, as the years passed the Dragoon became more and more cavalry related instead of infantry related, and at the end of its use it was the main cavalry, so a mounted unit that could also fight on foot rather then a infantry unit that could also fight on horse.
But for the Dragoon vs Knight I'd assume the Dragoon would be better off on horse then on foot. Maybe not though.. If he was light and really, really agile he could try and dodge the charge of the knight and either shoot or try to stab/slash the horse and try to dismount the knight for then to try and "out agile" him when they were both on foot.
I'm picturing the Dragoon preforming a "Indiana Jones Roll" to avoid the charge and then pwn the horse.
But still... A knight was a formidable fighter on foot as well.. So unless the Dragoon is able to shoot him I think the knight would win..
EDIT: @ Kasimir: Sorry, you posted while I was writing so I didn't catch your post right away.
The matchlock was not ignited by holding a match/torch, it was a really, really slow burning fuse that you lowered down into the gunpowderchamber of your gun. It was rather safe, but the fuse could twist a bit so you'd miss the hole. While on foot and with lots of time (IE at least a second or two) you could rearrange the fuse, but on horseback it would be harder.
As for your gunpowderpoach to explode/ignite during combat is rather improbable since you would have the pouch around 40-50 cm (around 1,5 feet) away from your fuse at the closest (IE having your pouch on your left hip or left side of saddle and your gun on your right eye, other way around if you had left mastereye), and when you reloaded your gun with powder you'd remove the fuse. You could have a small metal container some place easy to reach, like on your right side of saddle, or in front of you to place the burning fuse in. And, since the fuse was slow burning the risk of glowing debris from it would be rather low, so the gun would not go off due to this either.
It is mainly the poor accuracy and power that reduce the efficiency on horseback, not the risk of blowing yourself up.
As for the early use of hand guns.. The earliest use in rather large scale is dated to 1260, but it is not certain that they used handguns at that time, just some writings about it, not any "real" evidence. But along 1300+ the hand cannon was used. On that note, there is a sculpture from the 1100 displaying a weapon that looks like a hand cannon.
However, as said, the earliest guns was unable to take out heavy armour, but during 1500-1600 quite a lot happened. So, there could be an armourpiercing firearm that could kill a late heavy armoured knight.
Actually, it had to be, since the use of heavy armoured knights ended much because of heavy armour not being protective enough and therefor obsolete. So we can not say that the Dragoon would loose cause his guns could not penetrate armour.
So yes, it all depends on where the Dragoon hits, and if he hits at all...