As a DM I want to build an immersive and memorable, but also logically coherent world. I have a question about races.
What are the rules for mixing races?
For example, is it possible to be half elf half gnome (elf and gnome parents)? Can I mix everything with everything, or not, and why?
If so, what are the rules for racial traits? I want to use this to create some unusual NPCs, so racial traits are needed.
Since you asked for a worldbuilding answer as well, let's have a look at the default 5E setting of The Forgotten Realms. Here's a quick list of "Half-breeds" and how they work that I found in Realms-lore.
Offspring of an Elf and a Human. Look like a pretty straight fusion of "half human, half elf." It is noted on p.14 of the Forgotten Realms Campaign Setting (a 3E fluff book) that unions of elves and other races existed as well, but were very rare.
Much like a half-elf, this is typically the result of the union of a human and an orc--and looks like a tidy fusion of the two. The only other 'half' varieties of Orc I can find are an Ogrillon (half orc, half ogre) and a Tanarukk is the result of a orc and a fiend. There's some snatches of older lore that I can't remember which sourcebook it comes from that says Orcs are incredibly cross-fertile and can make a 'half' with almost anything...but I can't source that for you, so take it with a grain of salt.
These show up in the 5E Monster Manual, being the result of Human + Ogre. We don't really have a ton more information on them and, as mentioned above, the only known other pairing of Ogre + ? is with an Orc, having the formal name 'Ogrillon.'
According to Dwarves Deep, a 2nd Edition FR fluff book, a Half-Dwarf is the result of a Dwarf reproducing with a human, elf, gnome, or halfling. Curiously, it seems that Dwarven racial traits are extremely dominant--as a Half-dwarf is said to be nearly indistinguishable from a pure-blooded dwarf, apart from some variance in height and some relatively minor differences in their features (such as a half-elf/half-dwarf having pointier ears and a bit more angular of a face).
Additionally, if a Half-dwarf has a child with a Dwarf, you get a pure dwarf. If they have a child with anything else, you still get a half-dwarf.
Found only in the Forgotten Realms novel 'Pool of Radiance,' a Half-gnoll is Human + Gnoll. They are kinda horrifying. Said to be 'very rare.'
According to 'Realms of Infamy' a set of short-stories officially set in The Realms, a Half-goblin is what happens when you mix Goblin + Anything. In the story, we see that half-goblins derived from a human results in a taller goblin with eye colors in the human spectrum, rather than the standard goblinoid red.
Now on to some more exotic mixtures...
A stable population found in Dambrath resulting from a cocktail of human, wood elf, and drow, with all of their traits mixing together to get a unique variety of half-elves with a range of skin tones from ash-gray to a more 'human' peach. (Shining South, p.101).
What happens when a hag tries to reproduce and ends up with a male instead of the desired female. (Unapproachable East, p.12)
Offspring of an elf and a fiend (Races of Faerun, p.118)
This shows up in the 5E Monster Manual as a template that you can apply to literally any beast, giant, humanoid, or monstrosity.
Tieflings, Aasimar, and Genasi can be the result of cross-breeding between elementals, celestials, or fiends with humans. But it's also entirely possible that the cross-breeding happened many generations ago off several different family trees, and the recessive 'planar' traits just happened to pop back up in you.
So, technically the result of cross-breeding, but more often in the way that you may suddenly get great-grandma's hair color, even though it hasn't been seen in the family since her.
Of course, it's also possible to get 'plane-touched' due to other reasons, such as blessings, fiendish bargains, or other assorted weirdness.
There is plenty of lore-precedent for you introducing more varieties of half-breed into the world. Have fun! As another poster has suggested, I heartily recommend having a look at Tasha's Cauldron of Everything's Custom Lineage Rules for a guide on quick and easy racial homebrewing
There are no specific rules for mixing races in D&D other than the existing "half" races present in the Player's Handbook (Half-elf and half-orc).
Of course, there is nothing preventing it either if you, as a DM, decide that it is possible (an alternative would be to rule that certain combinations simply aren't viable).
This is getting slightly into home-brew territory but there are plenty of simple options.
This article presents plenty of options that should enable you to decide how mixed races are presented in your world.
On top of that, Tasha's Cauldron of Everything presents optional rules for customizing origins. Briefly - feel free to swap around the ability score improvements that races get to better represent their origin or background. E.g. if a gnome normally gets +2 Int and +1 Dexterity, then perhaps a half-gnome could, instead, get +1 Strength, +1 Int and +1 Constitution to represent the slightly larger stature that comes with a bit of human ancestry.
Skill and tool proficiencies could probably be swapped around as well without upsetting balance, though be wary of swapping about too many racial features and ending up with a character with the best of everything, making them more "powerful" than other races (though that may only be an issue for player characters; for an NPC anything goes!)
Tasha's Cauldron of Everything offers guidelines for realising Character Options, including
A way to customize your character’s origin by changing some of your racial traits
You may want to explore these options in the Customizing Your Origin section of the Character Options chapter. You can supply that mechanical guidance with conceptual guidance from Xanathar's Guide to Everything. Consider Chapter 1: This Is Your Life, and don't forget that you are looking at Ideas, not rules (XGtE, 61):
Even though these pages are full of tables [...] they don't make up a rules system - in fact, the opposite is true. You can use as much or as little of this material as you desire [...] For instance, you might not want these tables to help you decide who your parents [...] are, because that's among the information you've already come up with.
So while we have no strict rules that guide mixing races, we do have a lot of guidance that encourages you to realise character concepts with custom origins and supply them with reasonable backgrounds and how to balance racial traits.
If you want to deepen your understanding of racial trait options, I recommend reading the Detect Balance Sheet.
