Clucas, 26, has signed for four years for an undisclosed fee after passing a medical and agreeing personal terms.
The Swans had an earlier £12m bid for Clucas rejected.
He arrived as Swansea sought a replacement for Gylfi Sigurdsson after the Iceland midfielder's £45m move to Everton.
The new arrival could make his Swans debut when they seek a first win of the season at Crystal Palace on Saturday.
While Swansea survived in the Premier League last season, Clucas shone even as Hull were relegated.
Kingsley, 23, has signed a three-year deal with Hull, also for an undisclosed fee.
Before the deal was confirmed, Swansea manager Paul Clement underlined the player's versatility - he can play in defence, midfield, on the wing and was a striker while at Leicester's academy.
From there he went to the Glenn Hoddle Academy before rising to the top flight via Hereford, Mansfield and Chesterfield before joining Hull in 2015.
"He has experience of playing at Premier League level, he has a lot of experience of playing in British football," said Clement.
"And I also like the story, I like his background. He's had to fight to where he's got to through the route he's come in the lower levels.
"He was at the Glenn Hoddle Academy, he became a very strong and established player at Hull and I think he's going to be a really good addition to us."
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Swansea City have signed midfielder Sam Clucas from Hull with left-back Stephen Kingsley going to the KCOM Stadium in part exchange.