Is it possible to shoot the night sky?

I know astrophotography is a thing, although I've never tried it but, but does it work on video? What if I wanted to take shots of the milky way on an Ursa Mini, is that even possible?

I don't think you'd be happy with the amount of noise and the relative lack of detail. Most astrophotography requires large sensors to let in as much light as possible, while also using precision motors that can follow the stars while the shutter stays open for a long period of time.

No, it's not possible. The BMPCC can shoot only the brightest stars (And ISS), and if you try to sum a sequence of frames in order to improve the sensitivity then you get lots of FPN. Other BM cameras behave similarly.

I know astrophotography is a thing, although I've never tried it but, but does it work on video? What if I wanted to take shots of the milky way on an Ursa Mini, is that even possible?

Yes, It's possible - to some extent - with the URSA Mini, URSA Broadcast or URSA Mini Pro. You'd have to reduce the shutter speed and undercrank - which provides an altered sense of time (timelapse effect) - but it is possible. You'll also get the best results if you go somewhere with very little light pollution.

Most "video" of the night sky is done as a series of long exposure still images that are animated in an image sequence. Think more stop-motion rather than "video".

You can do this. What you'd need is:

A Schmidt or Maksutov Cassegrain telescope, if you're looking to do deep sky imaging (like closeup views of the Horse Head Nebula, Andromeda, Crab Nebula, etc).

The bigger the better, obviously; Mak-Cass 'scopes tend to be fairly compact relative to magnification, but it's a lot easier to find larger Schmidt-Cass 'scopes -- Mak-Cass 'scopes have an objective lens in addition to a mirror, Schmidts do not.

You'll need to devise a way to mount the camera to the eyepiece. There ARE adapters that you can acquire that basically hold a camera over the eyepiece; you'd focus the camera on the eyepiece, and then use the focuser to bring the sky image into focus. The ones I've seen are mostly geared toward dSLRs, so you might have to do some digging to find one that can handle an Ursa Mini.

You'll need a LOT of power and media.

You're definitely going to need a German Equatorial Mount that supports computer controls, and ideally that also includes a GPS. Some of these can calibrate themselves, some you need to calibrate when you place them. In some cases I think the software helps you with calibration. It's been a long time though, so the tools have probably gotten nicer since. There are some pretty amazing pieces of technology out there for finding deep sky objects.

Ideally, you'll want an autoguider. That's an imager that attaches to the finder and the computer, and the control software uses the image to dynamically correct drift.

When choosing the mount, make sure you pick one that's robust enough to handle the load of the OTA, camera, mounts, finder, autoguider (if you're using one), heaters (you want to keep the temperature of the OTA pretty constant to keep condensation from forming on the optics), eyepiece, counterweights, and so on... so a cheap Orion probably won't be robust enough plus you need one that can interface with control software for this to work... so you're probably looking at GEMs like Losmandy's. You'd be surprised at how much weight this adds up to. I know astronomers who ended up buying new cars just to transport their telescopes to places with dark skies.

Since these are night exposures, you can probably get away with a higher than usual ISO. The coolness will help to mitigate digital noise, but the technique I'm describing will also.

You'll probably end up recording in cDNG, but I suspect that braw with high quality settings would work just as well. Choose the slowest frame rate that you can, and set the shutter to 360 degrees.

Keep the lens as wide open as you can, and be very careful with focus. Most higher end telescopes have two-stage focusers to help with that. They're very precise.

Record for as long as you can. A lot of the really detailed images of objects like the Orion or Horse Head nebula are 6-8 hour exposures.

Now the fun part (and much less expensive).

Get a copy of something like Astrostack (which was free when I first used it, and probably still is).
It will probably require a collection of raster rather than raw image files, so you'll have to export your video as a sequence of still images.
Import the whole collection into Astrostack. It will crunch them down into a single image, simulating a single exposure that was approximately the length of your video clip.

Thanks for the information, guys! I'll give it a try.