That was the first one that came to my mind too.
For a recent example, if not necessarily epic, Extraction has several minutes of people running out of a car and into and around a crowded apartment complex while fleeing from a swarm of cops. Good editing to disguise the cuts, too.
Not from a movie, but there’s a great foot chase in an episode of The Rockford Files. Jim is meeting someone for a ransom exchange at the top of Hoover Dam. The Feds come and break up the party, but Jim has to chase the ringleader himself. So he does; all through the dam. They both come out the bottom of the dam just too tired to keep chasing anymore.
Amen. First, and only, one that came to my mind !
So, so good.
Wasn’t there a pretty good one in Baby Driver?
More McQueen:
This was a great sequence but I still can’t remember what it was:
Didn’t Chuck Norris have a shootout on top of the Chicago El in Code of Silence?
That was going to be my second suggestion, after Raiders. It’s mostly just Cage as the one on foot, but I love the cinematography in it.
Tell No One (2006), a French thriller where the protaganist (wrongly accused as often happens) is chased by cops, down stairs, over fences, then across a busy highway, dodging speeding cars and causing a huge pileup - cars, trucks, a motorcycle, almost getting creamed but somehow making it, and if a few innocnts are killed / hurt in the process, no matter. Later, running like a son-of-a-bitch, his foot slides on the pavement and he takes a huge fall - ow! (It’s a stuntman, but still pretty shocking.)
My favorite is the foot chase in “Men in Black” that gains the notice of Agent “K” and paves the way for Will Smith to become an agent himself.
For me, it’s Dangan Runner/Nonstop by Sabu.
Most of the movie is a chase on foot, and all of it is about the running or the people who are running.
Some sort of fight, anyway, according to TV Tropes (under the “Mood Whiplash” entry).
Yes, but the structure of the chase is very similar to Bullitt. They used some of the same set ups. There even was a VW.
Also, in no way could a 2-barrel 350 single exaust automatic transmission equipped “Pontiac Nova” sound like that. Not only did they use the Bullitt engine sounds, they even left in that weird “double clutch upshift” that Bullitt’s 4-speed Mustang did.
Not to mention, Bullitt didn’t let the bad guys get away. 
District B13
Basically the entire movie is just an excuse to show David Belle, the “inventor” of parkour, doing a bunch of cool parkour stuff.
A typical scene (NSFW language, but only if your coworkers speak French):
Tony Jaa from Ong Bak:
From a movie that might fall under the ‘Things you like but are not supposed to’ pigeonhole, ‘The Warriors’.
The scene in northern Central Park NYC where two members of the Warriors are being chased by the entire ‘Baseball Furies’ gang. Running flat out but cannot shake the Furies, and are getting winded to the point they can hardly speak. I can almost feel their tightness of chest and shinsplints…and the knowledge that if they’re run down they’ll be beaten to death with baseball bats.
Yeah but it’s got the shaker hood of the 1974 GTO. The only year they used the Ventura body for the GTO and the last year of the “real” GTO, which had a 350 4-barrel and dual exhaust. Was good for the low 16s at about 88-89 mph in the quarter mile. Much stronger than the 2BBL 350 but well within the strangulation era of “performance” back then…stock anyways. Could be a cobble-up job for the film. Quite possible given a high budget.
True enough about the phony manual trans sounds though.
Al Pacino during the big robbery scene in Heat: