‘You know I’ve applied for your passport, right?” the British DJ Charlie Sloth says to Drake after the rapper’s Fire in the Booth freestyle verse on BBC Radio 1Xtra – the joke being that the Canadian is over in the UK so much, and aligns himself with its music culture so strongly, that he might as well move here permanently.

For Fire in the Booth, MCs deliver long-form, often withering verses, free of the demands of song structure. And while Drake’s entry may not have the political ferocity of Akala’s, the determination of Wretch 32’s, or the spectacular endurance of Kano’s, he acquits himself well. His take may seem flat and incantatory to some, it is riveting when he spills over bar divides.

It followed another UK freestyle he delivered this month for the online channel Link Up TV. Over a light and ghostly drill beat, Drake dipped into UK slang such as “paigon” and “road” with a flow that nodded to London rappers such as Headie One and Skengdo, and an accent that veered between Jamaica, Brixton and his native Canada. Some listeners cringed: there’s a slight air of a trendy youth worker or, worse, a racist impersonator.

But like his Fire in the Booth, the freestyle became a sensation and Drake got away with that accent – his mixed-race heritage and background in cosmopolitan, West Indian-influenced Toronto means he can borrow from across cultures more authentically than other artists. “That’s how he talks – when you’re out with him … he’s saying ‘me and the mandem’,” Sloth tells me. “It’s not an act, it’s not Drake trying to fit in.”

More importantly, the two freestyles come off the back of years of British patronage. His 2010 debut album was American through and through, but after a UK tour that saw him watch Chelsea, eat at the Wolseley, and – very Drake, this – buy first editions of Charlotte’s Web and Winnie-the-Pooh, he started working with the Londoners Jamie xx for his second album, Take Care (2011). Another emotive neo-balladeer, Sampha, from south London, was featured on his third album, Nothing Was the Same (2013), along with the Scottish producer Hudson Mohawke.

At around the same time, Drake also started getting into British rap, naming the grime originator Wiley when asked about his favourite MCs, and then quoting the Tottenham MC Skepta on a track with Lil Wayne. His Instagram briefly became a digital shrine to grime rap battles, humbly saying the clash between Skepta and Devilman was “a true reminder that, whenever you think you’re good at what you do, there’s most likely two people out there that will yam your food”.

Drake has since taken the energy of these relatively small-scale spats and turned then into blockbuster sparring – his recent tete-a-tetes with Meek Mill and Pusha T have been two of the most acid and discussed beefs in hip-hop history. Skepta took credit in 2015: “That’s a real London, grime attitude to have. It’s good, it’s battle rapping … I definitely think that my vibe has rubbed off on Drake a bit.” For his part, Skepta opened up his track Shutdown with a Drake sample, and the pair got tattoos of each other’s crew symbols and joined forces for a track with the Nigerian pop star Wizkid.

Drake performing with Giggs at Reading festival, August 2017. Photograph: RMV/REX/Shutterstock/Rex/Shutterstock

Drake’s patronage started moving beyond grime into London’s road rap scene, hopping on stage with Section Boyz in 2016 for his own track Jumpman in a seamless segue. US hip-hop was always a component of grime’s syntax, but this performance underlined how blended both cultures had become, each with digital access to an entire world of rap as well as the styles on their respective doorsteps.

In 2017, alongside the Midlands soul singer Jorja Smith, Drake worked with the south London rapper Giggs on two tracks for More Life. One, KMT, became a moshpit anthem in the UK but was decried by DJ Akademiks in the US as “other people’s musical culture infiltrating, and kinda almost being forced upon US hip-hop culture” – a view that underlines how progressive Drake’s approach is in the context of a US rap scene that has long ignored the UK. “Unselfishly, he has provided a platform for British artists to excel,” Sloth says.

Drake has talked of his phone filling up with +44 codes on a remix for the young south London rapper Dave, while his wardrobe filled up with British “roadman” staple Stone Island. And he became such a fan of the black British drama Top Boy that he bought the rights. A series revival is coming up in 2019. As his Link Up freestyle shows, he has kept up to date with the recent drill sound, too, also namechecking Loski in an Instagram story hymning the influences on his new album Scorpion. “Drill is a genre that has had so much negative press, so to have the biggest superstar on the planet tip his hat to that art form … everyone was grateful,” Sloth says.

Some of Drake’s moves are canny, even cynical, but he clearly has a deep engagement with British culture, and it has been reciprocated: his latest pop track, In My Feelings, is likely to hit No 1 in the UK this week. But by also being an outsider in today’s radically porous and permissive London music scene, he has shown that anyone – through sheer respect and skill – can ascend to the top of it, even a Canadian who sounds as if he is from Camberwell.