Kingsman

Short Answer: India (and the rest of the former British Empire)

The Kingsman series has had storylines intersecting with major world events and figures, merging the elements of the Bond stories (elite British secret agents use high-tech gear to prevent major attacks) and the National Treasure series (purporting to show the "real" events behind historical milestones and figures-- the ones even those who failed middle-school history can name.)

But they also share a trait with a franchise whose one movie, sadly, prevented a movie series from developing: The Phantom. Like The Phantom, the King's Men existed over time, with various agents stepping in as "Galahad" and "Lancelot" as their predecessors are decommissioned or killed. 

It can be argued that the same is true for "James Bond"-- that the reason he is played by so many different actors is that the "James Bond" name is not that of a person but of a position, along the lines of Chief of Police... or, closer to our purposes, Green Lantern or The Dread Pirate Roberts.

In any case, my first concern was that the King's Men being not the department of a government, but a project funded by wealthy and noble families, was limiting. They would only be concerned with those situations that threatened British safety. Meanwhile, the British government sends Bond all over the world, because today's Britain is part of the global community.

Then I remembered that, oh, right-- the British Empire used to span the whole world anway. So, as long as we set the plot somewhere in the British Empire, we're fine. Time-wise, it has to be between WWI (when the third movie told us the project began) and now. But that gives us 100 years to work with.

And, since the first three movies were all about Europe (and a bit in the US [also a former British colony!]) the next stories should be set elsewhere in the Empire. And few locales are as different from Europe as India. 

We even have a name even history F-students will know-- Gandhi. Weird Al (in his "UHF" movie) wondered what Gandhi would be like in an action movie, but now we can really find out. Much of Gandhi's story plays out after WWI and he is not killed until 1948. 

The King's Man agents involved can be anyone who survived the third movie (a prequel) and but not introduced as a new recruit in the first movie. Which means we can have Harry back (played by someone else! Poor Michael Caine is 89 as of this writing. And he would have been much younger in the 1940s anyway).

The Kingsman series can continue with stories set today-- and continue to compete with Bond, Bourne, M:I,  and the other such action-plus-sci-fi-weapons franchises on that front. But as we have seen with the Indy series and the first Captain America movie (and its criminally under-seen Agent Carter spinoff), setting such stories in the past-- with steampunk and noir elements, plus period cars, sets, costumes, and music-- would really set the franchise apart in a thrilling way.

Now that Kingsman has gone into the past, maybe it should stay there awhile. But now that they have done the obvious and used Rasputin (like so many other franchises), please no Freemason plots (National Treasure did that to death). 

Instead, use the vastness of the British Empire to visit the Caribbean, the Pacific Islands, colonial Australia, even the stunning wilds of Canada. Not now, like Bond does, but back then. Between, more or less, 1920 and 1960.

And if they won't give us this, let's try The Phantom again.