When extracting part layouts from the full score in Dorico, if I display a cue in the full score not by using the popover but simply by copying and pasting and then shifting the octave, will instruments with different key signatures be automatically transposed?
If you’ve entered these as actual notes and then transposed (and probably changed the size), then these are just notes in the staff like any other, and they’ll be transposed just like any other.
What you lose is that if the source notation changes, the cue won’t change along with it.
If the key signature is different but I don’t transpose and just copy and paste while moving the octave, will it end up being different from the transposed position?
I’m not sure I understand your question. Are you asking what happens if you copy notes from Bb clarinet, for example, and paste them into the oboe staff? The pasted notes will retain their concert pitches, just like any notes that you copy and paste between staves. So a concert G, which is written as an A in the clarinet staff, will appear as a G in the oboe staff.
All of this is very easy to try for yourself, and you’ll see what happens!