|The title of this article is conjecture based on information revealed in the A Song of Ice and Fire novels or related material and may be subject to change.
The Kingdom of the Three Daughters entered into the great civil war in Westeros known as the Dance of the Dragons, on the side of the Greens, and engaged in the massive Battle of the Gullet against Rhaenyra Targaryen's navy near Dragonstone.
The alliance later dissolved, and all three Free Cities eventually went back to fighting each other over lands, trade, and wealth.
In the Books
The formal name of the triple alliance between these three Free Cities was actually just "The Triarchy" - which was a confusing misnomer for two reasons. First, Volantis was already famously ruled by a council of Triarchs - a "Triarchy" is a system of government, and calling itself "The Triarchy" was comparable to calling itself "The Kingdom". Second, the alliance was not actually ruled by such a council of three "triarchs" at all, but a council of 33 Magisters (apparently 11 from each city).
The formal name apparently didn't make much sense to people in Westeros, so they took to calling it by a popular nickname: the "Kingdom of the Three Daughters", because all nine of the Free Cities are commonly called the "daughters of Old Valyria" due to being founded as Valyrian colonies. Even this nickname, however, also introduced a misnomer, because it wasn't really a "kingdom" either (possessing no monarch as ruler). It is unclear why they didn't just call it "the triple alliance", etc. Many of their enemies in Westeros also frequently referred to it as just "The Three Whores".
The Kingdom of the Three Daughters was a short-lived alliance between three Free Cities which were formerly bitter rivals, only existing for a little over thirty years centered around the reign of King Viserys I Targaryen in Westeros. All of the Free Cities had fought each other in the Century of Blood after the Doom of Valyria, ending around the time of the Targaryen Conquest, but afterwards the fighting levels declined to a background level of the occasional limited trade war. In contrast, Lys and Myr remained bitter enemies in a near-permanent state of war, devastating the southwestern heel of Essos between them, which in time became known as the Disputed Lands as a result. Tyrosh often got dragged into these conflicts on one side or the other, as did Volantis, shifting from one to the other with the tides of war.
Around 96 AC, however, Lys and Myr surprisingly set aside their differences and united with Tyrosh in a triple alliance, realizing that with their combined forces they had the power to challenge even mighty Volantis. At the Battle of the Borderland (in the Disputed Lands) they shattered Volantis's armies and pushes its holdings back far to the east.
The unexpected political union of these three formal rivals into a larger realm upset the balance of power throughout the Free Cities, and even Westeros. With Volantis left reeling to the east, the Three Daughters turned their attentions northward, threatening Pentos along the border with Myr. Even Braavos saw its merchant fleet increasingly challenged by the growing power of the Three Daughters - though its influence was not so great that it could attempt to actually conquer Volantis, and further inland, Norvos and Qohor were little concerned by its growth. The greatest effect of this alliance was that the fleets of all three cities combined were finally able to crush the pirate-lords in the Stepstones who had carved out increasingly powerful kingdoms for themselves, using the rocky isles as bases to raid ships passing into or out of the Narrow Sea for fortunes in loot. The navies of the Three Daughters pushed westward to destroy the pirates in the Stepstones and then permanently conquer the islands outright.
The Three Daughters first created their alliance in the waning few years of the reign of King Jaehaerys I Targaryen in Westeros, who died in 101 AC and was succeeded by his grandson Viserys I. At first, the union of the Three Daughters was welcomed in Westeros, as it brought an end to the increasing pirate activity in the area. The Three Daughters' navies did levy high taxes on all passing ships in exchange for this protection, however, and as the years passed the taxes actually grew worse than what the losses to the pirates used to be.
The Iron Throne wasn't willing to go directly to war against the far-away powers of the Three Daughters, but as the three decades of Viserys I's reign passed, various factions fought proxy wars over the Stepstones. In 106 AC Corlys Velaryon and Prince Daemon Targaryen assembled an army of sellswords and younger sons to invade the western Stepstones: Lord Corlys supplying the large fleet of House Velaryon, and Damon acting as commander riding his fierce dragon Caraxes. Daemon pushed the Three Daughters out of most of the Stepstones and even crowned himself "King of the Stepstones", a rough and short-lived kingdom on those barren rocky islands formed from former pirate dens and his sellswords. By 110 AC, however, the Three Daughters counter-attacked by entering into an alliance with independent Dorne, who attacked from the west. Daemon's forces gradually lost ground, though he built up alliances with the Three Daughter's other rivals, Pentos to the north and Volantis to the east. In 115 AC Daemon's first wife died in the Vale so he returned to the royal court to pursue other political opportunities: having lost interest in the losing war in the Stepstones he did not return, and within a few years the Three Daughters had taken control over all of them again.
Years later in 129 AC, the great civil war within House Targaryen known as the Dance of the Dragons broke out, and Daemon Targaryen became one of the major leaders of his niece-wife Rhaenyra Targaryen's faction, known as the Blacks. The opposing faction of her half-brother Aegon II Targaryen, known as the Greens, did have many advantages at first - controlling the largest land armies, the three largest cities in Westeros, and the old royal treasury - but control of the sea was not one of them. The Royal Fleet under the Velaryons to the east had joined Rhaenyra, and soon the Iron Fleet of the Greyjoys sided with her as well. With no naval power on the east coast, Aegon II's Hand of the King, his grandfather Otto Hightower, prudently entered into an alliance with the Three Daughters against their mutual enemy. The Triarchy needed little convincing that if their old nemesis Daemon was allowed to control the Iron Throne it would spell disaster for their strategic interests. Dorne didn't join their union however: fighting one dragon in a proxy war on the Stepstones was one thing, but having been burned and ravaged by dragons before, the Dornish were wary of inviting their full wrath (that, and both the Reach and the Stormlands had declared for Aegon II, and they were Dorne's historical enemies).
One year into the war, the Three Daughters sent a massive combined fleet of 90 warships to attack Dragonstone and Driftmark. They engaged the Velaryon fleet and Rhaenyra's dragonriders off the coasts in the Battle of the Gullet, one of the largest naval battles in history. The engagement was a bloody and pyrrhic mess for both sides: one third of the Velaryon fleet was destroyed, and some Three Daughters ships slipped past their lines to raid and burn Driftmark itself, yet a full two-thirds of the Three Daughters fleet was destroyed before ultimately retreating back to the Free Cities. The worst blow to the Blacks, however, was that during the battle Rhaenyra's eldest son and heir Jacaerys Velaryon had been killed, along with his dragon Vermax, and for this reason the battle was nominally considered a Three Daughters victory.
Nonetheless, it appears that the Kingdom of the Three Daughters never recovered from the loses which its fleets took, and by the next year it was noted that it was "tearing itself apart", apparently as internal factions blamed each other or tried to seize on the opportunity to gain more power. Through as yet unclear events, one admiral in Lys's remaining fleet assassinated another in a palace intrigue, and the Triarchy dissolved into civil war. Seeing the opportunity to destroy the threat to their south for good, several of the northern Free Cities then entered into a brief counter-alliance to attack it: Pentos, Braavos, and even Lorath.
The "Kingdom of the Three Daughters" dissolved and when the dust settled, Tyrosh and Myr were left devastated. Lys, however, appears to have actually emerged positively from the breakup, and in the ensuing Regency era of Aegon III Targaryen after the Dance ended, it was said that Lys briefly became the most powerful of the Free Cities, surpassing even Braavos for a short time. The Rogare banking family of Lys even grew more powerful than the Iron Bank of Braavos itself, and ironically, entered into an alliance with Aegon III and the pro-Velaryon faction at the royal court. This period when Lys dominated the post-Triarchy political landscape in the Free Cities was known as the "Lyseni Spring", but it ultimately fell apart after a single generation due to other intrigues.