As someone who likes to run games with a mishmash of technology levels, my eyes went wide when I first got to the description of firearms in the DMG. One of my biggest fantasy influences is Stephen King's Dark Tower series, which blends post-apocalyptic sci-fi, high fantasy, and Westerns to tell a mind-bending epic across multiple worlds, and in which the main character is the last "Gunslinger," an order of profoundly deadly gunfighters who live by a strict code and basically never miss - that are basically the Knights of the Round Table of their lost culture.
As you might notice, living in the modern world, guns have basically replaced nearly every other type of personal weapon in our day and age, in large part because they're relatively easy to use (compared to a bow or a sword) and the damage they inflict at long range is very high.
That makes it a challenge in a game like D&D, where you still want people to be fighting with bows or with swords or other kinds of weapon.
The other RPG I've had the most experience with is World of Warcraft. WoW actually has guns as a very common weapon type, but they simply make guns equivalent to bows or crossbows - it's primarily an aesthetic choice (and, for about a decade now, only one class would use any of those sorts of weapons, meaning they're really basically indistinguishable on a mechanical basis).
D&D's statistics for firearms are different. But they're not in the base rules of the Player's Handbook, instead added in as part of the Dungeon Master's Guide as an option.
The problem is that guns are just... better.
Closest in style to the kinds of weapons found in the PHB are the "Renaissance era" firearms.
These are the Pistol and the Musket.
The Pistol does 1d10 piercing damage, and has the loading and ammunition properties, and has a range of 30/90.
If we compare this to the Hand Crossbow, we're up by two damage dice (meaning on average 2 more damage per hit) and we lack the light property and our long range is slightly shorter (90 instead of 120). But otherwise, the properties are the same.
The Musket does 1d12 piercing, has loading, two-handed, ammunition, and a range of 40/120.
If we compare this to the Heavy Crossbow, we do 1 higher damage die, and the range is admittedly far shorter (40/120 instead of 100/400) but it's also not a heavy weapon, so a Small player character can wield it.
The range issue I think is maybe a counterbalance to the Musket's power, but honestly, I think it's rare to have fights where you'll be able to make use of that 100-foot range.
The later "eras" of weapons go up in damage - and range - pretty quickly. All the "modern" firearms do two dice of damage. The lowest of these, the Automatic Pistol, does 2d6, which is the highest than any melee weapon can do, and then you get something like a Hunting Rifle doing 2d10. Futuristic weapons go even crazier, with the Anitmatter Rifle doing 6d8 necrotic damage. (Even if we consider that the exception, a Laser Rifle, which seems like it should be common in a sci-fi environment, does 3d8).
The point is, how does one balance this if you want to make these weapons available in-game?
Being able to shoot from range is a big advantage, and I think that's why we don't see things like Longbows or Heavy Crossbows every quite hitting the high damage that you can get from a Maul or Greatsword. But a Shotgun is already doing more than a Maul can, and you can be 30 feet away.
Apparently, in earlier editions, the solution here was the chance for a Misfire. You might notice that Matt Mercer's Gunslinger Fighter subclass (which was used by Taliesin Jaffe's Percy once they switched to 5th Edition D&D from Pathfinder when they started streaming) introduces this risk when shooting firearms. At base, any weapon will, on a Natural 1, suffer a misfire, which then requires the Gunslinger to try to un-jam the gun in the middle of combat. And if they fail, they need to do so over the course of a rest.
Thus, the high damage of the gun comes with a big risk.
In all honesty, this is probably the most reliable way to counter the power of firearms. In Mercer's breakdown of the subclass, different weapons have different Misfire ratings, meaning that even a natural 2 or higher might cause a misfire, and some of the subclass features can cause you to increase the misfire chance.
While flavorfully it makes more sense that weapons from more advanced ages would have a lower chance to misfire (as manufacturing grows more effective) I think you could potentially adjust the misfire chance to get higher the more damage you put out - indeed, you might say that the number of damage dice is the misfire chance.
Let's do some speculative napkin math to see how these wind up affecting the damage output.
We're going to look at four scenarios - in each case, we have a level 11 fighter (giving us 3 attacks per round) who is going to be using a two-handed ranged weapon. We're just going to say that a misfire means that they cannot make any more attacks that round, rather than the whole "try to repair" thing. PHB weapons have no misfire chance. Renaissance ones have a Misfire on a 1. Modern weapons have a misfire on a 1 or a 2, and the laser weapon in the Futuristic category will misfire on 1, 2, or 3.
We'll assume a +5 to Dex, and +5 proficiency. We'll also say we've got a +1 weapon, and we're shooting a target with an AC of 18. We're also going to assume the character has either the Crossbow Expert or Gunner feats, to ensure that we're not worrying about the reload feature. The reason I want to compare damage per attack, rather than per hit, is to make sure that we're accounting for the misfires.
This will grow a bit more complex, as the possibility for subsequent shots becomes lower the higher our misfire chance.
Heavy Crossbow:
With +11 to hit and no misfire chance, we have the following hit array:
Miss 1-6 (30%) Hit 7-19 (65%) Crit 20 (5%)
Hit Damage: 1d10+6, or 11.5
Crit Damage: 2d10+6, or 17
Total Damage per Attack: 7.475 + 0.85, or 8.325
Damage per Round (3 attacks): 24.975
Musket:
Now, we introduce a chance to misfire. This essentially means that we're going to need to multiply the damage per attack by 19/20 (or 95%) once to represent the average damage per second attack (accounting for the chance there is no second attack) and then again for the third attack.
Misfire 1 (5%) Miss 2-6 (25%) Hit 7-19 (65%) Crit 20 (5%)
Hit Damage: 1d12+6, or 12.5
Crit Damgage: 2d12+6, or 19
Total Damage First Attack: 8.125 + 0.95, or 9.075
Total Damage Second Attack: 8.62125
Total Damage Third Attack: 8.1901875
Total Damage per Round: 23.8864375
This is interesting: the chance to misfire actually winds up being a bigger detriment than the lower damage the crossbow is doing. The difference is relatively tiny, but I'd certainly count that as balanced to keep the firearm from being too powerful. But what about modern weapons?
Hunting Rifle:
Here, the damage of the weapon jumps up significantly - 2d10 is an average of 11, whereas the previous two weapons had an average damage of 5.5 and 6.5. However, we're also raising the chance for a misfire significantly - a 1/10 chance. Thus, each subsequent "damage per attack" is going to be only 90% of its previous potential. So let's see.
Misfire 1-2 (10%) Miss 3-6 (20%) Hit 7-19 (65%) Crit 20 (5%)
Hit Damage: 2d10+6, or 17
Crit Damage: 4d10+6, or 28
Total Damage First Attack: 11.05 + 1.4, or 12.45
Second Attack: 11.205
Third Attack: 10.0845
Total per Round: 33.7395
Ok, so here we've seen a massive jump in damage output, so even if there's now a one in ten chance to misfire, the higher damage makes it totally worth it - in contrast to the Musket, where we've actually made the Heavy Crossbow a better choice.
Laser Rifle:
In this case, the damage is going up again, but arguably by a less steep slope, which might make the 15% chance to misfire balance it closer to the Heavy Crossbow. In this case, each subsequent attack is only 85% of the previous one because of that chance to misfire. Let's take a look:
Misfire 1-3 (15%) Miss 4-6 (15%) Hit 7-19 (65%) Crit 20 (5%)
Hit Damage: 3d8+6, or 19.5
Crit Damage: 6d8+6, or 33
Total Damage First Attack: 12.675 + 1.65, or 14.325
Second Attack: 12.17625
Third Attack: 10.3498125
Total per Round: 36.8510625
I'll confess, I'm surprised this did better than the Hunting Rifle, because its damage wasn't that much higher than the modern weapon, and I'd have thought the misfire would be too big a deal.
However, I think misfire numbers could be tweaked, potentially, to balance these out. I'm tempted to try the Hunting Rifle at a misfire of 3 and maybe the laser rifle at a misfire of 4 or even 5.
Well, we've got time! Let's try:
Hunting Rifle (Misfire 3):
We can hold onto some of our earlier values. The first attack with the Hunting Rifle did an average of 12.45. But this time we'll multiply it by .85 instead of .9 for each subsequent attack.
Second Attack: 10.5825
Third Attack: 8.995125
Total: 32.027625
We're still almost ten damage per round beyond what the Heavy Crossbow and Musket were doing. What about Misfire 4?
Hunting Rifle (Misfire 4)
First attack: 12.45
Second Attack: 9.96
Third attack: 7.968
Total per Round: 30.378
So, we've gotten to a point where we're totally screwing up our weapon once every five shots, and it's still doing significantly more overall than the more archaic weapons.
Granted, this could very well be one of those "how it feels" versus "how it works" situations. Even if your overall damage output is going to be better with a high-misfire hunting rifle, the danger of misfiring might feel like a bigger risk overall than it really is.
Still, I do feel like maybe misfires aren't the best way to balance these weapons.
In Starfinder, ranged weapons don't add your Dexterity modifier to the damage. This means that melee weapons have a slight leg up, as you've always got that base damage modifier. Maybe this could be a way to keep their damage in check?
Given that we're going to try this instead of misfires, the math will be a lot simpler (though crits favor more dice rather than static bonuses). But let's get a basic idea here:
If you have a +3 to Dexterity, your nonmagical longbow will deal 1d8+3, which is an average of 7.5 damage.
If we don't get that bonus, a Musket's 1d12 does an average of 6.5 damage. Ok!
The Hunting Rifle does 2d10, which is an average of 11.
The Laser Rifle does 3d8, which is an average of 13.5.
Now, your Dex might go up to +5. In that case, the longbow is now doing 9.5 damage. It's outstripping the Musket, but not the modern or futuristic weapons.
In fact, even if we changed it so that a +X bonus on firearms only affected their chance to hit, and not the damage they dealt, a +3 Longbow would still only do an average of 12.5, while a Laser Rifle is outpacing that with no modification.
Essentially, what I'm learning here is that balancing these weapons is very tricky.
I think that perhaps the ultimate decision you've got to make is how important balance is to your campaign. If it is, the best solution might be to treat firearms as mere re-skins of the base weapons from the PHB - your Hand Crossbow is actually a Flintlock Pistol, and your Heavy Crossbow is actually a Rifle.
On the other hand, D&D doesn't really need to be balanced on the edge of a knife like some computer games are expected to be. These weapons could simply be more powerful, and just rarer.
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