The Eversmoking Bottle magic item has the following details :

Smoke leaks from the lead-stoppered mouth of this brass bottle, which weighs 1 pound. When you use an action to remove the stopper, a cloud of thick smoke pours out in a 60-foot radius from the bottle. The cloud's area is heavily obscured. Each minute the bottle remains open and within the cloud, the radius increases by 10 feet until it reaches its maximum radius of 120 feet. The cloud persists as long as the bottle is open. Closing the bottle requires you to speak its command word as an action. Once the bottle is closed, the cloud disperses after 10 minutes. A moderate wind (11 to 20 miles per hour) can also disperse the smoke after 1 minute, and a strong wind (21 or more miles per hour) can do so after 1 round.

What happens if a very fast creature opens the bottle, then immediately moves far away from the smoke (ex: 500ft) ?

1. The smoke follows the creature to the destination, but without leaving a trail on the way


2. Smoke is generated along the way as well, making a 60ft wide, 500ft long line of smoke


3. Smoke is only generated at the start of the creature's turns, so if he stops 500ft away and waits for his next turn, a single new 60ft cloud will be made there


4. The smoke is only generated once (upon opening time), no additional smoke will be generated neither on the way to the destination, or at the destination itself


5. Other behaviour

Option #4: The smoke only exists at the initial point of creation.

There are a few important points in your quote:

When you use an action to remove the stopper, a cloud of thick smoke pours out in a 60-foot radius from the bottle. [...] Each minute the bottle remains open and within the cloud, the radius increases by 10 feet until it reaches its maximum radius of 120 feet.

Note that other than the initial opening of the bottle, no other mention of the cloud's position refers to the bottle. Additionally, in order for the cloud to continue growing, the bottle must remain within its range.

While a "60-foot radius from the bottle" almost describes the fog as an aura that follows the bottle, it distinctly states that the bottle has to remain within the cloud for the full effect, so its implied that the cloud can be separated from the bottle.

If you were to move out of the cloud, the cloud would sit in place, cease to grow, and disappear when you cork the bottle (or by a stiff wind or the duration running out).

#4 The cloud remains where it was summoned but does not vanish (unless the requirements for dissipation are met).

First, the bottle only generates one cloud of smoke:

a cloud of thick smoke pours out in a 60-foot radius from the bottle

Next, the bottle must remain within that cloud to increase the cloud's size:

Each minute the bottle remains open and within the cloud, the radius increases by 10 feet until it reaches its maximum radius of 120 feet.

In real life, there would be a trail of smoke and one might consider that as being part of the same cloud, but the spell's description does not say "it continues to billow smoke" but that the cloud increases in size. It would be logical to interpret the cloud's increase in size as the result of more smoke being added to it from the bottle, but this is one of those times where a magical effect doesn't even come close to mirroring what you'd expect to happen in reality. It's magic (not physics) and a game (not a simulation)!

It's also moot, because once/as long as the bottle is outside of the cloud's area, the cloud stops growing but does not begin to disperse (unless there's a strong enough wind) because:

The cloud persists as long as the bottle is open [...] Once the bottle is closed, the cloud disperses after 10 minutes.

So in your scenario, the initial cloud of 60' radius is created, the bottle bearer rapidly moves out of its area, stopping the growth. No new cloud is created but the old one stays in place at the same size until the bottle is restoppered or a strong enough wind begins to dissipate the cloud.