The Shichibukai were introduced quite early into the One Piece series as seven powerful pirates who roam the seas freely. Unlike regular pirates, they weren't chased by the Navy because these pirates worked for the government and helped them eliminate other pirates. In return, they were allowed to continue living their lives on the seas with great freedom and continue being merciless pirates. Becoming a Shichibukai certainly isn't easy as it requires one to have tremendous strength and an aura that would intimidate other powerful pirates.
As the story went on, these seven pirates became crucial to the politics of the many islands that Luffy visited. In One Piece's Alabasta arc, Crocodile, one of the Shichibukai, was the main enemy and stood in Luffy's way. Similarly, the Thriller Bark arc saw Gecko Moria control the Florian Triangle from the shadows and steal shadows of everyone passing through the sea. The Shichibukai also ended up playing a massive role and the Paramount War, where the Whitebeard Pirates and their allies clashed against the World Government and these seven pirates. The Dressrosa arc was quite similar to the Alabasta arc and saw Doflamingo attempt to run a country into the ground. While the Shichibukai certainly were helpful to the Government, their abuse of this power eventually led to the dissolution of the system. Now, every single member of the Shichibukai is being hunted by the Navy, just like every other pirate.
Buggy's Fate
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Buggy rose to the position of Shichibukai high after the Paramount War, thanks to his connections to the likes of the Roger Pirates and Shanks. After being given this position of power, he established himself as a great figure in the underworld of the One Piece world. Following the dissolution of the system, Buggy was hunted just like any other pirate, however, he seems to have survived the ordeal. It was recently revealed that Buggy has continued to fall upwards and earned himself an even more prestigious position as a pirate. Buggy is one of the Yonko of the Sea, following the fall of Big Mom and Kaido. It is highly likely that he will play a crucial role in the arcs moving forward, given that he holds such a prestigious title. Buggy will continue to be hunted, however, the alliances that he has will certainly come to his aid and help him out when needed.
Mihawk's Fate
Dracule Mihawk is one of the very strongest characters in the One Piece world. As the World's Strongest Swordsman, Mihawk needs little to no protection and isn't afraid to sail the most perilous seas alone. While the Navy certainly is hunting Mihawk, he isn't one to shy away from a challenge. In chapter 956, it was quite evident that Mihawk relished the challenge from the Navy. He certainly won't be defeated before Zoro finally takes him on in battle and, as such, is in no immediate threat. At the same time, the Navy also likely realize that sending their full force after him would just be a waste as he is simply too strong and too dangerous to be dealt with so casually. Mihawk's role in the final saga isn't defined just yet, however, it'll likely be something very interesting.
Boa Hancock's Fate
Boa Hancock is one of the most beloved characters in the series and one who certainly fans are looking forward to seeing in action. Having been given the title of Shichibukai after just one voyage, it is no surprise that Hancock is feared all over the world. Following her being stripped of this position, the Navy have targeted Amazon Lily, however, it is likely that the island itself is safe.
While Hancock is strong enough to hold off any danger, Oda's message from Jump Festa from a few years ago seems to imply that she is in some sort of danger. It is possible for Hancock to have been captured by someone as capable as an Admiral, however, she most certainly is going to be rescued by someone like Luffy in the Final War. Fans will then likely get to see what she's truly capable of in battle and why she deserves the title of "Pirate Empress."
Edward Weevil's Fate
Weevil is one of the most curious additions that Oda has made to the story so far. Being introduced after the timeskip, Weevil is the self-proclaimed son of Whitebeard. While he certainly doesn't look like him, his strength resembles that of the former Yonko, as confirmed by Kizaru himself. He's supposed to be monstrously powerful as he has already wiped out a bunch of Whitebeard Pirates allies. It is possible for him to be captured by the Navy, however, that would serve no narrative purpose as he isn't someone around home a rescue arc can be based on. Weevil will most certainly survive the encounter against the Navy and then either fight against Marco or end up becoming an ally. While he does seem like an antagonist, it is his mother who's been manipulating him and there is a possibility that he could well become a good guy with time.