When Radiant Performer enters the battlefield, it triggers, copying Field of Ruin's ability for each other nonbasic land your opponents control.
Resolve the first Field of Ruin ability, destroying the land it it targeting, then causing each player to search their library for a basic land card and put it onto the battlefield.
River Song triggers for each opponent who searches this way.
Resolve the first trigger, putting a +1/+1 counter on it and dealing damage equal to its power to an opponent.
Repeat step 6 for each other trigger.
Repeat from step 4 for each other Field of Ruin ability.