Moses Boyd was born on April 30, 1991, and grew up in Catford, in south London. He describes himself as second-generation West Indian, with Dominican heritage on his father's side and Jamaican on his mother's. He began playing drums at Sedgehill School, at about the age of 13 according to AllMusic, and took lessons with a local teacher. He took part in the Tomorrow's Warriors jazz education programme and graduated from Trinity Laban Conservatoire of Music and Dance in 2016 with a Bachelor of Music degree in jazz drums. His early listening ran from Miles Davis's Kind of Blue to Dizzee Rascal's Boy in da Corner, a pairing that anticipates the mix of jazz and London club music in his own records.

In 2014 he won the Worshipful Company of Musicians' Jazz Young Musician Medal and made his first recordings. He played on the Peter Edwards Trio's Safe and Sound (March 2014) and drummed on "Move" on Zara McFarlane's If You Knew Her. That year he also formed a saxophone-and-drums duo with the tenor saxophonist Binker Golding, a fellow Tomorrow's Warriors alumnus. Their first album, Dem Ones, was recorded live straight to tape on October 9, 2014, and issued by Gearbox in 2015; it won the Best Jazz Act prize at the 2015 MOBO Awards. In 2015 he also received the PRS for Music Foundation's Steve Reid InNOVAtion Award and released Footsteps of Our Fathers, the first EP by his own band, Moses Boyd Exodus, whose line-up has included the tuba player Theon Cross, the guitarist Artie Zaitz, the trombonist Nathaniel Cross and Golding.

Binker and Moses took Jazz FM's Breakthrough Act and UK Jazz Act of the Year awards in 2016, along with a Parliamentary Jazz Award for newcomer of the year. Boyd's Exodus band issued the "Rye Lane Shuffle" 12-inch (April 2016) and the Time and Space EP (May 2016), and he received the John Peel Play More Jazz award at Gilles Peterson's Worldwide Awards. The duo's double album Journey to the Mountain of Forever (Gearbox, June 2, 2017), recorded live to tape in July 2016, added Evan Parker, Byron Wallen, the harpist Tori Handsley, the tabla player Sarathy Korwar and a second drummer, Yussef Dayes. Boyd's own EP Absolute Zero followed on August 4, 2017, on his Exodus label, and he won a second MOBO Best Jazz Act award that year, this time under his own name. In May 2017 he shared the drum stool with Femi Koleoso on Nubya Garcia's debut EP Nubya's 5ive.

In 2018 he played on three tracks of Sons of Kemet's Your Queen Is a Reptile (March 30, Impulse!), an album that was nominated for the Mercury Prize that year and featured Shabaka Hutchings. Alive in the East?, the duo's live set recorded at the Total Refreshment Centre in June 2017, came out on June 22, and the Exodus album Displaced Diaspora appeared on September 28 with a long guest list that included Nubya Garcia on bass clarinet, Joe Armon-Jones, Zara McFarlane, Terri Walker and Binker Golding. In 2019 he was a resident host on BBC Radio 1Xtra, and in mid-2020 he stood in several times for Gilles Peterson on BBC Radio 6 Music.

Dark Matter, released on February 14, 2020 on Exodus, was billed as his debut solo studio album. He produced and wrote all ten tracks and played drums on most of them, with Theon Cross, Armon-Jones, Garcia, Golding, Poppy Ajudha, Obongjayar and Nonku Phiri among the contributors. It reached number 2 on the UK jazz albums chart, won Boyd the AIM Independent Music Awards' UK Breakthrough of the Year prize, and was shortlisted for the 2020 Mercury Prize in July. Binker and Moses released Escape the Flames in December 2020 and Feeding the Machine in February 2022, the latter produced and engineered by Hugh Padgham with electronics from Max Luthert. In March 2022 the direct-to-lacquer session Four Sides to Every Story, with Art Themen, Greg Foat and Baldo Verdu, was released, and in August 2025 came Opening Time, a trio album with Foat and the bassist Jihad Darwish.

In March 2026 Tomorrow's Warriors named him one of its inaugural patrons. He returned as a leader with the single "Say Yeah" in July and with Songs for Sinners, released on October 30, 2026 by Exodus (with Beggars Music), his first album in six years. Reviewers describe a vocal-led, gospel-rooted record shaped by the church he grew up in, London sound-system culture and jazz improvisation, with vocals from Raelle, Shantéh, Amber-Simone and Chuck Strangers, and a family "kitchen choir" on "Say Yeah".

Boyd writes and plays at the point where London jazz meets the city's club music. His records draw on grime, broken beat, drum and bass, jungle, highlife, afrobeats, reggae, soca and electronica as well as the jazz tradition, and he has described the result as an extension of Black music and the diaspora. He builds tracks as a producer, putting programmed drums, brass arrangements and vocal features around his own kit playing, and NME described Dark Matter as a melting pot in which complex jazz rhythms share space with other styles. Besides Miles Davis, he names Dizzee Rascal's Boy in da Corner as a formative record, and he credits Gilles Peterson with showing him how to use jazz in club culture. The Guardian has called him a progenitor of the current London jazz scene, and AllMusic places him among its leading figures. His duo with Binker Golding, which Discogs describes as semi-free jazz, is a looser and more improvisational outlet, built around saxophone-and-drums pieces recorded live.