A couple of things; time and circumstances.

First we know there are many ways in which life on the planet can be eradicated that are not yet preventable by humans. Asteroid impact, solar flare, volcanic eruption, and climate change, while humans also can kill ourselves off with a large exchange of nuclear weapons. So if you plot the chance of our existing over time, as long as we stay a single planet species that plot heads to zero long before Voyager interacts with another star system.

Then we have Voyager, which will take 17 thousand years (and some change) to go one light year. Nearest star system is Alpha Centauri at just over 4 light years away so conservatively 80 thousand years before Voyager gets there (assuming it is on a path that takes it to Alpha Centauri)

Doing the math, the probability that humans exist in 80 Kyrs versus human ability to travel at a significant speed (say > 0.5c) between the stars vs human nature to retain artifacts when they can from historic events.

It is a thought exercise, which I doubt we'll live to see any affirmative answer to :-)

Voyager 1 is going 38,610 mph, 2 is slightly slower than that.

Andromeda, one of our closest galactic neighbors is 220,000 light years away.

Imagining that Voyager 1 is going in the right direction and ignoring that the Andromeda is going to collide with us regardless, that works out to 3,821,000,000 years to get there. And that's for a very very close galaxy, on an astronomical scale.

Pretty optimistic to think we'll be around on that time scale, but I suppose it could happen.

Aren't both Voyagers still gravitationally bound to our galaxy? They have escaped the sun but not milky way. In this case they will never leave our galaxy.

Good point. A quick google gives 537 km/s as the Milky Way's escape velocity, or 1,201,000 mph. Not going fast enough for that.

Moot point for Andromeda since that's coming here, but it'll never ever get to any other galaxies.

Voyager’s stated velocity is relative to the solar system, which imparts its own velocity relative to the Milky Way to Voyager. Another quick google suggests Sol is swinging around the galaxy at 200 km/s so Voyager’s 17 km/s is really nothing.

Also depends on whether Voyager's velocity is in a useful direction, it could just as easily be counterproductive. But not much difference either way.

But might they not travel through another star system in this galaxy before getting anywhere near another galaxy?

"Voyager 1 will eventually come within 1.6 light-years of the star Gliese 445 — in astronomical terms, this will be a near-miss. But don't hold your breath; the "encounter," for lack of a better term, won't occur for another 40,000 years.

After that, it gets a little difficult to predict Voyager's journey, as chaotic motions within the galaxy make accurate predictions of stellar movements challenging. But given the wide swaths of empty space between the stars, the spacecraft's visit near Gilese 445 is probably the closest that Voyager 1 will approach another star, ever."

https://www.space.com/39161-fate-of-the-voyager-spacecraft.h...

OP said star system. You’re quoting distances to another galaxy.

I’m responding to

>Is it necessarily true that our species will not exist when Voyager 1 and 2 fly through in a galaxy far far far away a long long time in the future?

Thanks for your answer btw, it was exactly what I was looking for as I didn't know the speeds of the Voyagers.