Born in Kinshasa, Zaire, Mulumbu joined Paris Saint-Germain (PSG) in 2004.
It was reported by the French media on 26 January 2009 that he joined Premier League team West Bromwich Albion on a trial basis. On 2 February 2009, the January transfer deadline day, he joined them on loan with a view to a permanent deal at the end of the season. Mulumbu's first term at The Hawthorns was interrupted by injuries and he had to wait until April to make his Premier League debut, coming on as a sub in a 2–2 draw at Portsmouth. He signed for West Brom permanently on a one-year contract for a fee of £175,000 on 10 July 2009.
He was named West Brom's Player of the Year by both the club and the fans at the end of the season, and signed his third contract extension in 13 months on 25 July 2011, tying himself to the club until 2015 (with the option of an extra year).
On 15 March 2014, Mulumbu scored a crucial winner for West Brom in a 2–1 victory against Swansea City at the Liberty Stadium, giving Pepe Mel his first win as head coach.
In June 2015, it was announced that Mulumbu would sign on a free-transfer for newly promoted Norwich City on 1 July 2015.
On 3 August 2015, Norwich announced that Mulumbu suffered a broken metatarsal in a friendly against Brentford which would require surgery and sideline him for several weeks, ruling him out of the start of the season. He would finally make his debut on 27 October in a Capital One Cup game against Everton, which Norwich lost on penalties after a 1–1 draw. He made his Premier League debut a few days later on 31 October in a 2–1 defeat to Manchester City . However, with Jonny Howson, Alexander Tettey and Graham Dorrans preferred in the centre of midfield, Mulumbu would ultimately be confined to only five league starts and two appearances as a substitute in the 2015–16 season, plus one appearance in the FA Cup, and the season would end with Norwich relegated back to the Championship.
Mulumbu remained with Norwich for the 2016–17 season but again found his opportunities for first team action restricted. He would ultimately make only seven league starts and six substitute appearances, along with two appearances in the EFL Cup. His last appearance for Norwich came on 11 February 2017 in a 5–1 win over Nottingham Forest. On 2 May, it was announced that Mulumbu was one of seven players who would be released by Norwich when their contracts expired at the end of the season.
He was formerly a member of the France under-20 and under-21 teams. He earned his first cap for Congo DR national football team against Algeria on 26 March 2008. In 2009, Mulumbu decided to boycott international football after he claimed his national team lacked professionalism. On 23 September 2011 he retired from international duties, quoting "numerous problems we were facing with some authorities within the DR Congo Football Federation". However, in July 2012, he decided to revoke this decision, and made himself available for the Congo DR national football team once again.
Mulumbu is a devout Muslim.As of match played 26 December 2015