Either conquered or joined her willingly. If the Kyoshi Warriors are still around, who knows what they were doing. Maybe Kuvira didn't want them, thinking their methods were out of date. Or maybe not, who knows?
So, out of extreme curiosity to Kuvira's take over and Kyoshi Island, I began to ponder of two maps. One of the Avatar World with a few of the major stops along the series marked down, and another of the one Kuvira kept with her on her train. After tedious research that lasted a whole of three minutes at most, I have come to the conclusion that Kyoshi was never taken under Kuvira's control. Between the two maps, the island where Kyoshi should be on Kuvira's map does not exist. Of course, I'm 99% sure that Kyoshi Island was not obliterated or anything of the like, so it is my best guess that Kuvira either doesn't care about Kyoshi, or it has once again become forgotten
The island looked almost sunk anyway, probably not much land left.
It could be that Kuvira left Kyoshi Island alone for political reasons.
The island is confirmed to have become part of the Earth Empire, though it was not specified if it joined willingly or, like Omashu and Zaofu, resisted and was forcibly taken. As for the Kyoshi Warriors, they were likely press-ganged into the army or imprisoned if they refused to join.
Can I just ask how you know the island was confirmed to have become part of the earth empire?
It's on the Kyoshi Island article. Sorry, should have made that clear.
While this wiki is far better than most I've seen, I feel I should be the bad guy and mention that no Wiki is ever, on its own merit, an authoritative/official source.
As to the question of Kyoshi Island's status, yes, the article states it was taken, but the question one must ask is: What's the source? As is usually the case on this Wiki, fortunately the article has a citation and lists its source: Enemy at the Gates
While Kyoshi Island is not referred to by name specifically in the episode according to the transcript, the opening narration does specifically state the following:
That tells us that of all the individual city-states that made up the Earth Kingdom, Zaofu and ONLY Zaofu remains unconquered.
Thus, Kyoshi Island is indirectly but officially confirmed to have become part of the Earth Empire - though the how remains unknown.
2) The Kyoshi Warriors are either: serving in Kuviras military(they might have been forced when she took over, I doubt they would volunteer), or more likely, at the island, at least playing lip service to Kuvira. If Kyoshi Warriors can't stand against Azula, I doubt they could stand against Kuvira. If Kyoshi is free, their probably just hanging out on Kyoshi. Some might have volunteered to join the military of Kuvira to help reunite the empire, but no evidence of this has been shown.
NOTE:all of this is from my conjecture and deductive reasoning, I have no actual basis on sources for my guesses(while I do draw the guesses from fact).
Sorry, but last time I checked Suki was able to hold her own with Azula during the battle we saw. I have seen tons of people saying Kuvira is amazingly better than Azula, but if they were both in their prime Azula wouldn't be a pushover like we saw in her first battle with Korra.