In Karjakin's First Encounters with Carlsen, I noted that both Magnus Carlsen and Sergey Karjakin played in the 2004 FIDE Knockout World Championship in Tripoli, Libya, and that both were eliminated in the first round. This was their first of many participations in World Championship events. Let's look at their respective records since then.
Here I'll use a bit of shorthand to document the record. The notation 'C21' is a convention I developed for my World Chess Championship index to differentiate events played in (sometimes overlapping) FIDE cycles. The links are to my page on the 2004 Knockout tournament.
In the next cycle, the knockout event became the World Cup, a qualifying step to the rest of the cycle. Karjakin was again eliminated in the first round. Carlsen survived to the fourth round, where he lost the match but continued to play for a qualifying place in the subsequent candidates matches. He succeeded, but in the first match was eliminated from the rest of the cycle by Aronian.
In the next cycle, both players survived to the semifinal round of the World Cup. Carlsen was eliminated by Kamsky, and Karjakin was eliminated by Shirov. Kamsky beat Shirov in the final round to qualify for a subsequent match.
The next cycle saw the introduction of the Grand Prix series. Carlsen dropped out after playing one event and declined to play in the World Cup. Karjakin played in both, but was again eliminated in the semifinal round of the World Cup, losing to Gelfand, who went on to win the event.
In the next cycle, Carlsen was seeded into the Candidates Tournament by rating and narrowly won on tiebreak ahead of Kramnik. This catapulted him into a title match with Anand, which he also won. Karjakin was eliminated in the third round of the World Cup by Judit Polgar.
As World Champion, Carlsen was exempt from qualifying the stages of the next cycles. Karjakin qualified by rating into the 2014 Candidates. He finished second behind Anand, but after a minus score in the first half, he was never in the running for first. In the current cycle, he won the World Cup, then won the Candidates event, setting up a title match with Carlsen later this year.