The Tigers put in an offer for the 26-year-old restricted free agent on Monday with the Power having until Thursday afternoon to decide whether to match it or let him walk.
Port opted against matching it, with the club citing Chaplin's recent knee surgery as a key factor in their decision.
"Troy needed serious knee surgery towards the end of 2012 and while we hope he has many years of football left, we're not prepared to commit to the extent Richmond is," Port football manager Peter Rohde said.
"We would have liked Troy to remain at Alberton, but on terms suitable to the Port Adelaide Football Club as well as the player.
"In fact, we had what we considered a fair contract in front of Troy and his management for a long period this year, but unfortunately they were unwilling to sign.
"Troy has been a valuable player at Port Adelaide throughout his time with us and we sincerely thank him for his efforts with our club and wish him the very best at Richmond and back home in Victoria."
Chaplin has played 140 games since his debut in 2004.
He becomes the second free agent to end up at Richmond after former Crow Chris Knights joined the Tigers on Monday.
Melbourne have added some experience to their list with former Geelong midfielder Byrnes joining the club.
Byrnes, 28, has reportedly agreed to a two-year deal at the Demons and as an unrestricted free agent is free to move to the club of his choice.
He managed 108 games since making his debut in 2004 and was a member of the Cats' 2007 and '09 premiership teams.
But he fell out of favour in recent years, managing just nine games in his past two seasons.
Byrnes finished his career at Geelong on a high after playing a part in the club's VFL premiership this year.
"We thank Shannon for all he has given for the Geelong Football Club," Geelong general manager of football Neil Balme said.
"Shannon has recognised the opportunity to continue his career with Melbourne and we support him in his wishes.
"Shannon's efforts for the club were second to none and went a long way to achieving premiership success."
Meanwhile, Jared Rivers looks set to head in the opposite direction with the Cats the front runners to secure the defender's signature.
The 27-year-old is also an unrestricted free agent, although Geelong may put their trade targets in North Melbourne ruckman Hamish McIntosh and Gold Coast midfielder Josh Caddy ahead of Rivers' signature.The Demons themselves look set to be a big player once the trade period gets underway on Monday with the club linked to Collingwood duo Chris Dawes and Sharrod Wellingham.