Frodo and the Fellowship's journey across Middle-Earth on the quest to destroy the Ring introduces the audience to a plethora of people groups and places, and explores the various different races across the world. Tolkien introduces Hobbits, Elves, Dwarves, Ents, Orcs, Wizards, and even human Men, as well as multiple different subgroups of each. This world is so fully realized that even people and places that aren't explicitly part of the main storyline of Lord of the Rings still have backstories and their fates can often be found in Tolkien's supplemental writing.
Tolkien wrote The Hobbit prior to The Lord of the Rings, and it's odd that someone who spent a whole story with the race of Dwarves made them fairly absent from the events of the Lord of the Rings trilogy. Gimli is obviously a major character in the story, but he's basically the only Dwarf that ever impacts the story in a meaningful way. The part of the story that takes place in the mines of Moria is one of the only other places that Dwarves are even really mentioned in a wider sense. Different groups of Elves, Hobbits, and Men all make major appearances in the story, so why were the Dwarves left out? What was going on with the rest of them while the events of Lord of the Rings were transpiring?
By this point, many of the Dwarven kingdoms had fallen, and the ones that remained were scattered, inhabiting places like the Iron Hills, the Blue Mountains, and the Lonely Mountain (of Hobbit fame). But it's not as though these Dwarves were spending their time sitting around doing nothing; they were busy fighting their own battles for the War of the Ring. It's easy to forget that the fight against Sauron's forces was not exclusive to the places and people that the Fellowship encountered (like Rohan and Gondor), but instead was happening all over Middle-Earth.
The later parts of the Return of the King book (including the Appendices) mention what was going on with the Dwarves at this time. The Dwarves that lived in the Lonely Mountain, along with the Elves of Mirkwood and the army of Dale, actually fought against an army of Orcs that was advancing on them from the North. This is actually one of the reasons that the events of The Hobbit happened in the first place. Gandalf feared what Sauron would do if he were able to gain control of Smaug when he began the War of the Ring, and so Gandalf sent Thorin Oakenshield on the quest to reclaim Erebor in order to stop this threat before it happened.
There were conflicts like this happening during the War of the Ring, but the main story doesn't focus on them because the main focal point of Lord of the Rings is about the world of Men, and how their kingdoms were being expanded and revitalized while many of the other races (such as Elves and Dwarves) are fading away. Tolkien had already focused on the realm of Dwarves plenty during The Hobbit, and likely wanted to explore a different aspect of Middle-Earth.
Speaking of The Hobbit, what became of the Dwarves that were featured in that book specifically? Thorin, Kili, and Fili, of course, died during the Battle of the Five Armies at the end of The Hobbit. As for the other members of that party, Balin led a group of Dwarves from the Lonely Mountain (including Oin and Ori, who were part of Thorin's Company) to recolonize the mines of Moria. Balin was killed by an orc after five years of this colony existing, and the attacks only grew from there. Oin tried to escape the Orcs in Moria but was killed by the Watcher in the Water (the same tentacled creature that attacked the Fellowship in Lord of the Rings). Ori tried to make a final stand with the Dwarves around Balin's tomb, but they were killed, and the notes he left were found on his body by the Fellowship as they pass through Moria.
Another member of the Company, Gloin, was Gimli's father, and accompanied him at the Council of Elrond. He later died of old age in Erebor. As for the rest of the Dwarves from The Hobbit, including Dwalin, Dori, Nori, Bifur, Bofur, and Bombur, not much is known about them, other than that most of them also died of old age in Erebor. Dwalin actually ended up having the longest lifespan of any Dwarf, living for 340 years (the average life expectancy of Dwarves was about 250 years).
As for the main Dwarven character from Lord of the Rings (Gimli), after the War, he led a group of Dwarves from the Lonely Mountain to found a new colony in the Glittering Caves of Aglarond. These were a group of caves behind Helm's Deep that Gimli found during the events of The Two Towers book and became fascinated by. He became a Dwarf-lord, and since this new domain was in Rohan and close to Gondor, he was able to stay close with Aragorn, Eomer, and Legolas, who had himself taken a group of Elves to found a new colony in Ithilien. After one hundred and sixty years, when Aragorn finally passed on, Gimli and Legolas sailed over the sea to Tol Eressea.