How does the Flash move through space?

The Flash cannot move through interstellar or intergalactic space. In the instance mentioned by the original post, the "track" is provided by the alien extraterrestrials sponsoring the race. The Speed Force is not responsible for the Flash being able to run in space as is often mistakenly thought. The Flash is limited to running on a surface.

In the story called The Human Race (Flash #136 to #141) humanity is threatened by cosmic beings of immense capacity. An alien racer lands on Earth dying from exhaustion leaving only a cryptic message about being "too slow." The threat arrives in the form of cosmic, chair-bound aliens whose designs resemble Marvel's Celestials.

• Presumably for entertainment (since they called it gambling) they take members of alien species who have the capacity to move at incredible speeds and have them race against other species.

• The losers planets are summarily destroyed.

• At the start of the race, the aliens inform us: "We will construct a track through the entire structure of your space-time continuum. You will run. Communication with your homeworld will be made possible by instantaneous transfer of information through the fourth dimension. Our technology does not operate across distance."

• After entering a dimensional apeture, the Flash is off and running on the "curving fabric of space-time" as provided by the extraterrestrial gamblers.

Like this: The above excerpt from Wednesday Comics shows a few different mechanisms Flash can use to travel through Space-Time... time travel, vibration, vortices, and running. Flash, by and large, is not governed by mechanics... or it might be fair to say, he isn't universally governed my mechanics consistently. He often will invoke a physical principle or mechanic to his benefit in one area or respect, but ignore it in whole or in part in another area, arena, aspect, or respect.

The wind vortices are a good example... when he needs them to provide thrust to push himself back (and up) to slow his descent, they work to do that... but, conversely, if he needs them to generate a forwards wind while he's planted on the ground, suddenly that thrust effect on his person is ignored. The wind only blows forwards towards the enemy, fire, or whatever... but he doesn't get blown or thrown back.

This arbitrary ignoring of consequences is the basis of many of Flash's techniques like the Infinite Mass Punch, running faster than escape velocity, eating copious calories, etc.

What's important to understand then is that while speed is the lens by which Flash understands his powers and abilities, in truth it is a thematic physics-breaking Speed Force magic that is his true ability. The Speed Force was shamelessly lifted from the Star Wars concept of The Force... an ambiguous, magical, quasi-mystical justification for all the powers and abilities that make your sci-fi space heroes what you really wanted them to be: wizards and knights. Likewise, the Speed Force is an ambiguous, quasi-mystical justification for all the powers and abilities that make your sci-fi superhero what you really want them to be all along: a mortal manifestation of Mercury, the speedster god.

There is no particular reason that Mercury should run instead of fly or teleport... as a fictional god those constraints are arbitrary, but they're built into the fabric of our mythology and culture and create the expectation of what a speedster is or should be. So the Speed Force is an arbitrary literary construction that just gives Flash all the justification to be like Mercury without admitting he's a modern-day magical god. Understanding this explains why most mechanical explanations of Flash ultimately fall short because, under-girding it all, is the desire to make sure he fits that imprint.

Even if cherry picking all his powers across his comic history suggest he should be able to hit stronger than Superman (JLA), travel through time at will in a blink, think faster and smarter than Batman (DC One Million), freeze people in place like Medusa (speed steal), be immortal (JLA), etc. he will always, eventually, reset to the Mercury archetype which is a more powerful pull upon the character than the concerns of technical, scientific continuity.