By default, guns aren't a thing in D&D, with the technology levels almost being based around the notion of pre-dating firearms.
Historically, one of the key advantages and changes that came about with firearms was the way that they negated the effect of armor. Indeed, it may have been a democratizing influence on warfare (though the longbow certainly had had a similar effect far earlier) as the heavy, plate armor that rich nobles could afford had previously rendered them mostly impervious to the simple spears the peasant infantry would use could be easily punctured with a bullet.
Granted, a longbow or heavy crossbow could probably also pierce plate if it hit directly, and a musketball hitting at a shallow angle could probably be deflected by plate, meaning that the notion of plate continuing to increase AC even against powerful projectiles isn't entirely outside of the realm of believability.
In a world where literal magic exists, of course, all bets are off.
But D&D does come up against a tough balance issue: it seems obvious that firearms ought to do more damage than bows and crossbows. There is also a very clear advantage to using ranged attacks, as you can do so from a safe distance if the target is incapable of firing anything back.
The way things are balanced, the very highest regular ranged weapon damage you can accomplish in D&D 5E is the Heavy Crossbow, which deals 1d10 piercing damage on a hit (plus Dex.) Having the Loading property is its downside, though an Artificer with the Repeating Shot infusion or anyone with the Crossbow Expert feat can ignore that. (The infusion also means never running out of ammo, which is pretty nice.)
The best melee weapon damage is the Greatsword or Maul, which both deal 2d6 damage. The average then comes out to 7 (or 8.33 if you have the Great Weapon Fighting fighting style) compared to the Heavy Crossbow's 5.5.
However, a couple things can come into play. Setting aside bonuses like a weapon that adds additional die rolls (like a Flame Tongue,) ranged weapons have an advantage which is that you can have both a magic weapon and magic ammunition. If you have a +3 Heavy Crossbow and some +3 Bolts, that gives you a total bonus in damage (and chance to hit, which is not insignificant) of 6, compared to a max of 3 for your Maul. The extra three damage makes up for the d10's lower average damage than 2d6 (even with the fighting style) not to mention that you're also hitting more frequently thanks to the significant boost in your to-hit rolls.
Now, granted, we're talking primarily about a classless character with out any significant class features that could alter this math like Improved Divine Smite or a Barbarian's Rage.
But if we look at, say, a Champion Fighter, this actually means that, with access to magic ammunition and weapons, they're actually going to be dealing more damage at range than they could in melee.
And that's before we even get to firearms.
Again, the highest damage ranged weapons by default are 1d10, or 1d8 for a longbow, which doesn't require using a bonus action to reload or another feature to negate that requirement.
Firearms, naturally, need to be more powerful than these to make any sense. Pistols (by which I think the game means flintlock pistols like what you'd see a pirate shooting) do 1d10 damage. Muskets deal 1d12. At that point you're catching up quite easily with melee weapons even before considering magic ammunition - a Musket is doing 6.5 average damage.
And that's just where things start. There are three more categories: modern firearms start being two-die weapons. Automatic pistols do 2d6. Automatic rifles, revolvers, and shotguns do 2d8, and Hunting Rifles do 2d10.
Finally, when you get into futuristic weapons, you get 3-die weapons, dealing 3d6 with a laser pistol, 3d8 with a laser rifle, and then the insane antimatter rifle which does 6d8 - oh, and none of these are physical damage.
Now, sure, if you're going to have laser rifles, they ought to be dealing way more damage. But how, then, do you introduce those to a campaign where your Barbarian, Paladin, or Strength-based Fighter isn't feeling like they made the wrong choice?
One option is to look to Star Wars.
Star Wars is, hands down, the ultimate science-fantasy property, and they managed to come up with a way to let people fight with swords in a world where there were all sorts of blasters and guns.
But how do we represent awesome sci-fi melee weapons like a lightsaber? We could step things up like the futuristic firearms did by adding more dice. If a laser rifle can deal 3d8 on a hit, compared with a longbow doing 1d8, maybe a lightsaber can deal 4d6, compared with a greatsword doing 2d6.
The consequence of this, of course, is that everyone's doing more damage, which means fights are faster and deadlier. That might work for your campaign (though I think 5E combat is actually pretty quick already, at least in-world if not in realtime.)
Another option is far simpler:
Ignore the DMG and just reskin crossbows as guns. Re-skinning is usually a good option if you don't want to upset the game's balance. Just let players buy a Musket for the same cost as a Heavy Crossbow, make it do 1d10 damage and let them buy bullets for the same price they'd buy bolts.
Another idea is one that I suggested in an earlier post:
Maybe firearms resist magical enchantment. So while a Musket or a Hunting Rifle might do a lot more damage than your Heavy Crossbow would, it's a lot easier to find a +2 Heavy Crossbow and some Bolts of Dragon Slaying than it would to be find equivalent guns and bullets. The downside to this, as I see it, is that you introduce an exciting and unusual element into your fantasy world and then pretty much tell the players not to play with the fun new toys.
I'd love to see Wizards introduce some Weird West or Steampunk official setting, and perhaps flesh out rules surrounding firearms, but I also know that a lot of fantasy purists hate the idea of guns in their fantasy worlds (mind you, I'm no fan of guns in the real world. But I'm so influenced by fantasy worlds that include more modern weapons, like the Dark Tower, World of Warcraft, and, apparently, Final Fantasy VII, that I find the medieval stasis embraced by most fantasy pretty limiting.)
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