They were outclassed in the first set as Ukraine's Kichenok and New Zealand's Routliffe, who have won three Grand Slam doubles titles between them, raced to a 4-1 lead before taking the set with another break three games later.
Raducanu and Boulter came from 4-1 down to level at 4-4 in the second set.
Their best chance to turn the match around came in the following game, but they failed to convert four break points.
Despite saving three match points three games later, they succumbed to defeat.
Raducanu faces Rebecca Sramkova in the last 16 on Thursday and Boulter takes on fifth seed Diana Shnaider.