If you copy from or to a transposing instrument (and the score is set as a transposed instead of a concert score) Finale will display the notes in such a way that they reflect the same concert pitch.

As you suspect (and appear to be taking into account), when copying from one key into another, Finale will transpose the music into the new key (unless that has been disabled in the preferences/document options. Use the Key Signature Tool to be sure you are not copying from C major into A minor, which would of course show identical key signatures.

Peter, if you say both your source and destination file are in the key of C, yet copying chords give you something a minor 3rd lower. I suspect that one of your files is actually set to the key of A minor, not C major.

Thanks for the input. That was indeed the problem. The source was C major, destination A minor. Setting the destination to C major also took care of the second problem.