But there is no immediate word on Lance Franklin's controversial bid to leave Hawthorn to join Sydney.
White will leave Richmond to join the Power on a three-year deal, while the Dockers have also lodged the paperwork for Melbourne's Sylvia.
White and Sylvia are unrestricted free agents, meaning their former clubs cannot block the moves.
Sylvia's move was academic, with Melbourne and Fremantle confirming it on Thursday.
White, 26, has played 105 games for the Tigers and was named their most improved player this season.
He played 16 games this season, but missed the elimination final loss to Carlton because of injury.
Port Adelaide said in a statement that White attracted interest from several other clubs.
"It's a massive opportunity to come to a great club in Port Adelaide and I'm really looking forward to what the next few years can bring the club and me personally," White said.
"It's a massive thing for me to come over to SA and into a group that's played in two finals this year.
"I know Ken (coach Ken Hinkley) runs a pretty tight ship and I'm really excited about the challenge of trying to slot into the 22 each week and hopefully be playing finals come the end of next year."
The AFL has been looking into Sydney's controversial nine-year offer to Franklin, said to be worth $10 million.
Rival clubs are furious at the monster proposal and want the league to scrap the Swans' cost of living allowance in the salary cap.