The brands were previously with Wieden & Kennedy. The agency won the Finish account in 2014 from Havas. Nurofen was also with Havas before moving to W&K in 2015.
Wieden & Kennedy’s work on Finish includes the "Who cleans the cleaner?" campaign which brought together some disgusting combinations that end up in the dishwasher.
A spokeswoman for RB confirmed the account move, and explained that the business "review[s] agencies as a regular part of managing our business".
Helen Andrews, managing director at Wieden & Kennedy London, said: "We understand RB's change of strategy to devolve more control to local markets. We are immensely proud of all the work we created for RB, including the award-winning Finish campaigns, and wish them all the best in the future."