Franklin, who has played three games for the Swans this season, will sit out the Tigers clash after being unable to train during the week, per Sydney's Head of Physiotherapy and Medical Services, Damian Raper.
"Lance completed the game against Port Adelaide but suffered a knock to the knee in the last quarter," Raper said.
"He has been unable to train this week due to his knee soreness and will not play against Richmond this weekend."
Tom was sidelined in the second term of the loss to Port Adelaide after a heavy knock and will miss at least this week, while Paddy, who's innocuous head knock saw him immediately substituted from the game, is expected to be sidelined indefinitely.
The 26-year-old made his debut for the Swans in 2022 after stepping away from St Kilda in 2019 due to a string of concussions.
"Paddy is doing well in the early stages of his recovery," Raper said.
"He still has some residual symptoms which are not unexpected. He will be monitored closely in accordance with the AFL's HIA protocols.
"Tom is also doing well in his recovery but will miss the match against Richmond after entering the AFL's HIA protocols. He will continue to be monitored closely."
Sydney also confirmed that Sam Reid's recovery has suffered a setback, now not expected to return until after the mid-season bye.
The 31-year-old is yet to play this season as he recovers from a glute injury, now unable to run for a month due to a hamstring issue.
"Sam's rehabilitation has failed to progress over the last fortnight due to pain in his hamstring tendon," Raper said.
"After further investigations and consultations with external specialist, Sam will now not run for 4 weeks and we don't expect him to return to play before the midseason bye."
In better news for the Swans, Ryan Clarke is expected to be available for selection this weekend after returning to full training.