Dragonborn are the native race of my setting, in a way similar to Elves: Dragon gods had a massive fight, and wherever their blood landed a dragonborn rose. Though the evil dragon god won, the good dragon god's final action before death was to seal his foe away inside the planet. This is where volcanic reactions, tidal waves, and earthquakes come from. Though they are the original race, they aren't the most common after other races either came into existence (Halflings, humans, wood elves, dwarves, goblinoids) or migrated here while the planar bridges were established (All other elves, gnomes, tieflings, other Volo's Guide races.)
Tieflings and Aasimar aren't Fiend/Celestial blooded, and are instead the equivalent of "Humans" in their home plane that they have long since been cut off from. Demons, devils and angels exist here, but are few in numbers and have yet to truly establish new demiplanes, so fiend worshiping cults are small.
Goblinoinds are more dominant than humanoids, with Hobgoblins at the forefront of their kind. Eons of wars between the gods of humans and hobgoblins (and by extension, between the races themselves) have left the humanoid pantheon but a shadow of its former glory (With lesser and demi-gods forced to step up and take over for the fallen dieties) while the hobgoblins shift their focus to dwarves. The hobgoblin gods also have taken gods from other pantheons, including the Dwarven forge god (Known as the Traitor God among dwarves), the Goblin fertility goddess (The Hobgoblin god thought he had killed the goblin trickster and hunting god, then took the fertility goddess as a concubine/adviser. The trickster god fakes his own death, then revived the hunting god purely because he knew the hunting god would want revenge and thought it would be cool to see another fight between them), and the Human god of Knowledge and Magic (He foresaw the fall of the human pantheon and tried to warn them, only for the human gods to ignore his warnings. He offered his service to the Hobgoblin god in exchange for an end of hostilities for a time, and also acts as a double agent through various good aligned cults)
The age of man has long since passed, and all that remains is a handful of fractured city-states and settlements. Human culture is still as diverse as ever, but only in small groups, and nowhere near as influential. You'll also find many humans integrated within hobgoblin societies, running the gamut of the lowest slaves to the highest elected official. You still won't find humans in the highest of Hobgoblin society, but many hobgoblins would rather focus on their conquest of the world in the name of their god than the day-to-day of some cities within their nations.
Halflings are still closely tied to humans, but have more of a wanderlust than in Forgotten Realms. Halflings are more or less the "Gypsy" race, with everyone within six degrees of everyone else and a passing knowledge of someone (or at least someone who would know) of another tribe. Still, you'll find plenty of Halfling settlements near human ones or away from anyplace one would deem "Key to a military strategy".
Dwarves are still proud tradesmen, merchants, and soldiers, but aren't as tied to mountains as they are in Forgotten Realms. Sure, there are still mountain strongholds and such, but you're as likely to find wooden citadels in the forests, admiralty governed maritime islands (Some with a little Polynesian flair), and stone-carved fortresses in the highlands.
Wood Elves are the perpetual enemies of Bugbears (I'm thinking of coming up with a half wood-elf/half-bugbear race that's the result of attempts of marriage-induced peace, bugbear raids, or particularly rebellious elves and bugbears running off together), High and Drow elves still hate each other. High elves look at other races the same way Joseph Stalin looked at his own people, Drow look at them the same way most humans look at beef cows, and wood elves... kinda tolerate other races so long as they don't mess with their territory.
Gnomes are to Elves as Halflings are to Humans, only you won't find independent gnome settlements so much as "the gnome part of an elven settlement". Elves even tend to tolerate gnomes more than ant other race.
Orcs were once several tribes working on their own, but now you're more likely to see them either A: Integrated in human or hobgoblin settlements, or B: Part of the Great Host, which is a nomadic orc nation that hopes to beat the Hobgoblins in their conquest of the world. Unlike hobgoblin nations, however, Orcs seek to dominate and enslave all other races, where Hobgoblins are willing to work with other races to see their goals through. You'd still be able to find independent tribes of orcs, but when the Great Host comes they have two choices: Join or die.
Gnolls aren't tied to demons, but are still absolute nutcases.
All other races are more or less the same between the source books and my setting. (That, and I'm still working on the world, so I haven't gotten that far.)