I want to convert ProRes RAW files (.mov extension) to an open source format such as CinemaDNG or Blackmagic RAW. Is it possible? If so, how?
I want to advice against converting to any RAW format, I don't even know if that's possible at all. A RAW format is usually driven by the needs of the camera hardware and fits the purpose of efficiently storing the video data while providing very high flexibility, often much more than non-RAW codecs do. That's why you usually cannot convert to such a format, you can only convert from a RAW format to something else.
Regarding CinemaDNG, there is this unofficial information on Adobe community forums (Source):
The CinemaDNG format was dropped due to licensing issues and disputes. And only a few cameras were built to use it to begin with, so sadly support for that format was never great and is now nearly non-existent.
That said, it is best to avoid CinemaDNG.
But what should you do now? If I understand you correctly you need a codec for video editing, but ProRes doesn't work for you. There are a number of other options that are suitable as an high-quality intermediate codec (which means a codec that is suitable for doing video editing). Examples are Cineform, DNxHR or DPX. As far as I know, only Cineform is fully open source. Here's a table with codecs and their supported frame sizes, bit-depths and a few other key information: link
In order to transcode the ProRes file to your target codec, you need a tool that supports reading the first and writing the second one. It may be a little difficult to find something that can read ProRes RAW if you're on Windows, but you can install Apple's ProRes RAW. Beware, due to Apple's restrictive licensing politics, only the Adobe products listed there are officially supported on Windows. Other tools may be able to read the format too, but there is no guarantee that they correctly interpret the files. Which means, you should use Media Encoder or Premiere Pro to transcode the format to something else. In Premiere you can go to project settings-> ingest settings and choose another format. If you're unhappy with the choices add your own ingest preset. I have not yet worked with ProRes RAW in Premiere or Media Encoder on Windows, so I can't tell you if everything works well. But you can use a trial version of Premiere Pro or Media Encoder and test if this works for you. Also, Premiere Rush is supported too, but I've got no experience with that whatsoever.
On a Mac you can use Final Cut Pro, Motion, or Compressor to export ProResRAW to something else (see bottom of this article).
Note: Blackmagic RAW is not open source, there is no public source code available. They have a binary library/API which allows for decoding Blackmagic RAW in a host application.
Blackmagic RAW and ProRes RAW are both closed source codecs which provide video frames to the host application in a demosaic’d format - therefore neither of them can output something that can be re-encoded as RAW.
Very few RAW formats provide the ability to decode the data in such a way (mosaic’s in camera raw space) that can then be used as in the input to another RAW format. CinemaDNG is the only format which can do this due to being open source. There was once a tool that could read CinemaDNG and convert it to CineForm RAW (Both are open source so this could still exist)
Regarding CinemaDNG, it is a mature and well supported format (Adobe Premiere, Apple Compressor, DaVinci Resolve) and the only fully implemented true open source RAW format. It is also the only RAW video format which can be lossless.