Despite being named in the 2013 and 2014 Vodacom Cup squads, Kleyn never made an appearance, suffering a knee injury in 2013 and an ankle injury in 2014. He was included in the squad for the 2014 Super Rugby season. Despite missing the early part of the season with his ankle injury, Kleyn recovered to be named in the starting line-up for their match against the in Pretoria, but ended on the losing side as the home team won 28–12. A further three appearances off the bench followed in May 2014. In the second half of 2014, Kleyn was a key player for in the 2014 Currie Cup Premier Division, starting ten matches during the season. The final two of his starts came in their 31–23 semi-final victory over the and the final, in which Kleyn played the entire 80 minutes, helping his side to a 19–16 victory over the to win his first senior trophy. Kleyn established himself as a regular in the Stormers squad during the 2015 Super Rugby season, appearing in thirteen of their seventeen matches, including five starts. He helped the Stormers finish top of the South African Conference, and also played in their qualifier against the, a match won by the Australian visitors to eliminate the Stormers from the competition. He once again was a key player for a Western Province team attempting to retain their Currie Cup title in the 2015 competition, playing in all twelve matches, of which he started eleven. The team finished third on the log and Kleyn started their 23–18 victory over the in the semi-final and the final, where his side could not emulate their 2014 result, losing 24–32 to the. Kleyn suffered an arm injury in a warm-up match against the prior to the 2016 Super Rugby season, which ruled him out of the start of the competition. During the international break, Kleyn started two matches for Western Province in the 2016 Currie Cup qualification series, the first of those being a match against the during which Kleyn scored his first try in senior rugby, scoring ten minutes from time to help his side to a 43–34 victory. He was included in the Stormers touring squad for their trip to Australia to face the and the, but failed to be included in their matchday squad. He eventually made his only appearance of the season in their 52–24 victory over the in their final match of the regular season.
Munster
Kleyn secured an early release from his contract with in July 2016, leaving South Africa to join Irish Pro14 side Munster on a three-year contract prior to the 2016–17 Pro12 season. On 3 September 2016, Kleyn made his competitive debut for Munster when he came on as a substitute against Scarlets in the sides opening 2016–17 Pro12 fixture. On 17 September 2016, Kleyn scored his first try for Munster in their 20–16 away win against Welsh side Dragons in Rodney Parade. The match was also his first start for the province. On 17 February 2017, it was announced that Kleyn had been ruled out for 8–10 weeks due to a neck injury. The injury was sustained during the Pro14 fixture against Dragons on 10 February. On 1 September 2017, in what was his return to competitive action following the aforementioned neck injury, Kleyn scored two tries in Munster's win against Benetton in Round 1 of the 2017–18 Pro14. Kleyn scored a try and won the Man-of-the-Match award in Munster's 21–16 win against Ospreys on 16 September 2017. Kleyn was nominated for the 2018 Munster RugbyPlayer of the Year award in April 2018. He signed a three-year contract extension with Munster in September 2018, a deal that will see Kleyn remain with the province until at least June 2022. Kleyn was ruled out for six weeks after undergoing surgery for a thumb injury in November 2018. He returned from the injury in Munster's Pro14 fixture against Ulster on 21 December 2018. Kleyn earned his 50th cap for Munster in their 26–17 win against Leinster on 29 December 2018.
Ireland
Kleyn qualified through residency to play for Ireland on 8 August 2019, and was selected in Joe Schmidt's 44–man training squad for the 2019 Rugby World Cup. He made his debut for Ireland in their 2019 Rugby World Cup warm-up match against Italy on 10 August 2019, starting in the 29–10 win, and was selected in the 31-man Ireland squad for the 2019 Rugby World Cup.
