Each part in its proper key

See:
https://musescore.org/en/handbook/3/transposition#transposing-instrumen…

Why would you want that? The keys in your screenshot are already correct: if the globak key is C major, then B-flat instruments have D major key signature (written pitch is a whole tone above concert pitch), and E-flat instruments have A major key signature (written pitch is a minor third below concert pitch). No need to change the key signatures to flats at all, unless you want to change the global key, then just attach a global key and don't change the local keys.

To be clear: if you aren't aware, some of the instruments the instruments you mention are all transposing instruments, meaning the music has to be transposed for each of them individually. If you want to enter the music all in the same key for simplicity, that's fine, just make sure the "Concert Pitch" button is enabled. Then all the keys will match and all notes will be at their sounding pitch. Then when you're done, turn Concert Pitch off so MuseScore can automatically transpose the music as it needs to be when played on those instruments.

I think you make a big mistake, the Key signature is not the key of "fabrication" of the instrument. You want B flat instruments with B flat in key signature , so, 2 flats, you want Eb instruments with Eb key so 3 flats. If it is really want you want, it's completly wrong.

The key signature doesn't show that, it shows the tonality of the song you work in, it's always changes. In your case your song is in C (tonality of the C instruments) so Bb instr. are in D , and Eb instr are in A, all is correct. But if you change the tonality for C# you will see that all instruments change also, of course.