Stars in the Milky War Galaxy orbit the center of mass of the Mikly Way Galaxy, sort of like planets in a star system orbiting the central star.
The Sun orbits about 26,000 light years from the center of the galaxy, and thus about half way from the center of the galaxy to the outer edge of the galaxtic disc.
Since the Galactic disc is about a thousand light years thick, the stars in the disc form a rather flat disc like system, much like the planets in a typical solar system. The difference is that there are tens or hundreds of billions of stars in the galactic disc and that the typical distance between a star and its nearest neighbors is only a tiny fraction of the diameter of its orbit around the center of the galaxy.
One would expect that the Sun would spend half of its approximately 225 to 250 million year orbit "above" the mathematical central plane of the galactic disc and half of it "below" that mathematical plane. But the Sun sort of wobbles "up" and "down" around the central plane of the galactic disc over a period of millions of years which is much shorter than half of it orbital period.
The orbits of stars around the center of the galaxy are not perfect elliptical orbits. The stars in the neighborhood of the Sun have similar distances from the galactic center (whch is necessary to be in the neighborhood of the Sun), and similar orbital speeds, and similar orbital excentricities, etc. And the word is similar, not identical. Over thousands and millions of the years, small orbital differences can result in vast changes in the relative positions of two stars.
At the present time the Alpha centauri system is the closest system to the Sun, at 4.3 light years. Here is a link to a list of the 26 stars calculated to pass with 5 light years of the Sun within the last 3 million years in the past and the next 3 million years in the future.
and the closer two star systems pass to each other, the more their mutual gravitational attraction alters their orbits. So the Sun's orbit is constantly being slightly changed by passing stars, as well as sometimes by passing more massive objects like molecular clouds or star clusters.
And when did the aliens in "The Chase" leave their information in the DNA of lifeforms on different planets?
LAFORGE: This is not a natural design. Captain. This is part of an algorithm, coded at the molecular level.
PICARD: An algorithm? Are you saying that these DNA fragments are elements in some kind of computer programme?
LAFORGE: I know how it sounds, but there's no way this could be a random formation. This is definitely part of a programme.
CRUSHER: This fragment has been part of every DNA strand on Earth since life began there, and the other fragments are just as old. Someone must have written this programme over four billion years ago.
PICARD: So, four billion years ago someone scattered this genetic material into the primordial soup of at least nineteen different planets across the galaxy?
DATA: The genetic information must have been incorporated into the earliest lifeforms on these planets, and then passed down through each generation.
So the Sun and the Earth have orbited around the center of the galaxy at least 16 times since the DNA sequences were seeded, and the planets that the DNA was seeded on have spread out considerably from their original relative positions.
PICARD: Doctor, programme the computer to analyse the distribution of the pieces that we have, correcting for changes in star configurations over four billion years, then extrapolate for the missing piece.
CRUSHER: That's going to take several hours to set up and to process. If you'll excuse me.
Adn they are able to calculate the position of the planet with the DNA needed to complete the program.
So how large a region of the galaxy are the DNA segments spread over?
There is a scene where Professor Galen, expecting Picard to coming with him, moves his finger over a space map to outline their itinerary.
GALEN: The Vulcan ship will take us as far as DS Four. An Al-Leyan transport is scheduled to arrive at the station three weeks later. They'll take us as far as Caere, and then we'll use the shuttle to get us to Indri Eight, our first stop.
And considering the size of the galactic disc on the map, it looks like the distances of each leg of the trip will be thousands or tens of thousands of light years, and that the various planets with the DNA sequences are scattered over most of the galaxy.
However, it is my theory that the galactic disc on the screen is not to scale with the distances between the various stars shown, but instead is merely a scenic backdrop to the star map Considering how many decades it would take to travel the distances bypassed by the Bajoran Wormhole in DS9, or to return over the 70,000 light years that the Voyager was transported by the Caretaker in Voyager, the distances traveled in "The Chase" should have been only tens or hundreds of light years, not thousands or tens of thousands of light years.
So the available data is quite contradictory about how large an area of the galaxy that specific DNA program was seeded in.
Did the aliens seed other, presumably identical, DNA computer programs on planets in tens, hundreds, or thousands of other regions of the Milky Way Galaxy?
The message of the humanoid in the program says:
HUMANOID: You're wondering who we are, why we have done this, how it has come that I stand before you, the image of a being from so long ago. Life evolved on my planet before all others in this part of the galaxy. We left our world, explored the stars, and found none like ourselves. Our civilisation thrived for ages, but what is the life of one race, compared to the vast stretches of cosmic time? We knew that one day we would be gone, that nothing of us would survive. So, we left you. Our scientists seeded the primordial oceans of many worlds, where life was in its infancy. The seed codes directed your evolution toward a physical form resembling ours. This body you see before you, which is, of course, shaped as yours is shaped, for you are the end result. The seed codes also contained this message, which we scattered in fragments on many different worlds. It was our hope that you would have to come together in fellowship and companionship to hear this message. And if you can see and hear me, our hope has been fulfilled. You are a monument, not to our greatness, but to our existence. That was our wish, that you too would know life, and would keep alive our memory. There is something of us in each of you, and so, something of you in each other. Remember us.
Note that the humanoid says that life evolved on their planet before any other planet in "This part of the galaxy". It doesn't say whether they explored only part of the galaxy or all of the galaxy in their quest for other intelligent life. If the aliens had warp drive as fast as TNG era warp, it should have taken them less than a century to reach every part of the galactic disc and seed their DNA coad on various planets in each part of the galaxy, so they could have created many different copies of their program in many different parts of the galaxy, if they decided to.