Last week Nome put up his post about QoL changes to Thresh due to the problems related to his passive in Dominion.
He and other champs with abilities that give bonuses that depend on farming/last-hitting minions (Ashe, Cho, GP, Karthus, Nasus, Sion, Swain, TF, Thresh and Veigar) are naturally at a disadvantage in Dominion because efficient farming is not really possible in this game mode.
My solution centers around the one thing these champions could reliably get over the course of a game: Health Relics.
Why not make Health Relics give them an additional bonus equivalent to them last-hitting a certain number of minions?
Swain, Karthus and Cho could get additional mana (and health in Cho's case).
Ashe, GP and TF could get bonus gold for picking up the Relic (in TF's case any team member picking up the Relic would get it).
Nasus, Sion, Thresh and Veigar would get their permanent stat increases.
The exact number of minion last-hits a Health Relic would give obviously needs to be balanced out and could also be different for each individual champion.
I think this solution is better than having the same champions with different mechanics on SR and Dom.