This is more of a photography question than a video question. If you attempt this with a video camera you will need to blend, if you use a still camera you will have much greater resolution and ramping options.
There are many ways to do it, here's the best and shortest video I could find with a half hours effort: "Turn Your Star Trails Photos into a Video".
On the right are numerous links to other methods. It's technical and lengthy to explain each method. Some general tips are available here: "Getting Started With Star Trail Photography", "Instructables - Star Trails - A Beginner's Guide" and "StarCircleAcademy - Star Trails".
A short explanation is to use an intervalometer, increase your bulb time for each shot in aperture priority mode. That will give you 'moving dots' that stretch into long star trails if you stack them in post - exactly like in your example. Just take all your photos and make them into a video or an animated .GIF file.
If you are in the northern hemisphere then pointing your camera towards (<1° off) the North Star (Polaris is a binary star, α UMi Aa is at a declination of +89° 15′ 50.8″) produces circular trails> In the southern hemisphere you would point towards Sigma Octantis. Pointing towards the equator provides long streaks instead of circular trails. A longer exposure will produce longer star trails, but will also usually dim the brightness of the trails.
The actual location of the celestial poles varies due to axial precession. Because of a phenomenon known as the precession of the equinoxes, the poles trace out circles on the celestial sphere, with a period of about 25,700 years. The Earth's axis is also subject to other complex motions which cause the celestial poles to shift slightly over cycles of varying lengths; see nutation, polar motion and axial tilt. Finally, over very long periods the positions of the stars themselves change, because of the stars' proper motions. The apparent positions of the stars also change slightly because of stellar parallax effects.
When pointing at a celestial pole the stars complete one full rotation in less than 24 hours, and move almost 15 degrees every hour. In 24 Earth hours they travel 361°. Some easy math is available here: "Greg Boratyn - Night Photography".
If you want to shoot for more than 8 hours (sunset to sunrise) and for multiple days you might want to read: "Difference between sidereal day and solar day on Earth".
"Earth moves a little less than a degree around the Sun during the time it takes for 1 full axial rotation. So, for the Sun to appear on the same meridian in the sky again after 1 full axial rotation, the Earth has to rotate one extra degree to bring the Sun into the same apparent meridian in the sky. This is why the solar day is longer than the sidereal day by about 4 minutes.".
For a lengthy, almost easy to understand, explanation of nearly everything check out Guy Cook's webpage: "What we see in the Sky: Stars". To find magnetic north for your location try the "NGDC Mobile Declination Calculator", it uses the World Magnetic Model to adjust for anomalies in the Earth's magnetic field.
If you don't point towards the celestial pole but instead point towards Polaris you'll end up with a photo (from "Stellar Neophyte Astronomy Blog") like this:
The bright line is Polaris, the arrow points to the celestial pole.
Back to the simple explanation: Start a short bulb exposure, maybe 30 - 60 seconds, then ramp it up to an exposure time of 5 minutes and keep doubling it until you get a good result. Once you get close to the right exposure time, use your judgement and previous results to decide whether to increase or decrease the exposure time.
Some helpful software to cheat and blend what you have into what you want:
Use the "Exposure Triangle" to adjust between brightness, streak length and 'trail decay'.
Need more info just ask, I'm willing to make a longer answer when I have more time available. Check out some of the videos and find an example of what you want.
Here's a video of one of the better effects - stars move slowly and then streak, sometimes they unstreak other times the scene fades. Looking at the background gives tips on the interval of the blends. Well done work translates into any language, he speaks English too, but the description text is in Chinese.