The first works set in such a location are A Story of the Days To Come (1897) and When The Sleeper Wakes (1899) written by H. G. Wells, which take place in a future London. Thirty years later, the silent film Metropolis (1927) depicted several scenes showing moving sidewalks and escalators between skyscrapers at high levels. Later The Roads Must Roll (1940), written by Robert A. Heinlein, depicts the risk of a transportation strike in a society based on similar-speed sidewalks. The novel is part of the Future History saga, and takes place in 1976. Isaac Asimov, in the novel The Caves of Steel (1954) and its sequels in the Robot Series, uses similar enormous underground cities with a similar sidewalk system. The period described is about the year 3000.

In each of these cases there is a massive network of parallel moving belts, the inner ones faster. Passengers are screened from wind, and there are chairs and even shops on the belt. In the Heinlein work the fast lane runs at 100 mi/h (160 km/h), and the first "mechanical road" was built in 1960 between Cincinnati and Cleveland. The relative speed of two adjacent belts is 5 mi/h (8 km/h)[5] (in the book the fast lane stops, and the second lane keeps running at 95 mi/h (152 km/h)). In the Wells and Asimov works there are more steps in the speed scale and the speeds are less extreme.

The description above is from a "Moving walkway" article on Wikipedia.

I've always thought the concept of a network of variable speed moving walkways was a smart, exciting idea. For various reasons, it has not yet been implemented in any significant way. Partly, I think, because the technology does not exist that would make it economically feasible or operationally practical.

In this article from NewScientist, we learn about a few early, and other more recent, attempts to develop at least rudimentary systems of multiple speed sidewalks:

The first moving walkway had been unveiled [in 1894] at the Chicago World's Fair and had proved a huge success at subsequent expositions in Berlin and Paris. Chicago's walkway, the brainchild of engineer Max Schmidt, consisted of three rings, the first stationary, the second moving at 4 kilometres per hour and the third at 8 km/h, an arrangement that allowed walkers to adjust to each speed before moving to the next.

With the Brooklyn Bridge walkway, Schmidt upped the ante. This time he envisaged a loop system at each end of the bridge, with a series of four ever-faster walkways. Passengers moved from one to another until finally taking a seat on the benches aboard the fastest, which whisked them across the bridge at 16 km/h. Because the system ran constantly, there would be no waiting and little momentum lost on stops and starts.

In fact, the idea of high-speed walkways had been established in New York longer than anywhere else. Back in 1871, local wine merchant Alfred Speer patented the first "endless-travelling sidewalk", and promptly proposed an ambitious elevated moving walkway along Broadway. It would have zipped pedestrians along at up to 30 km/h, a prospect with comic possibilities that delighted pundits. One newspaper suggested that getting trapped with interminable bores would be a thing of the past: one "has only to suddenly step on the passing sidewalk to be carried rapidly beyond sight or hearing of his tormentor". Despite building a working model and lobbying state and city politicians for a decade, Speer discovered his invention was simply too visionary to find a backer.

Most likely we won't get a fully functional major implementation of this idea until molecular manufacturing is developed. After that, if "utility fog" can ever be perfected, we might see truly remarkable personal transportation systems that will turn science fiction into reality.