Why can't I import private key?

This question is from somebody who is essentially ignorant of how publi/private key encryption works - even after reading for a long time!

I’m using the latets beta release GPG3win (about GPA says GPA 0.9.1-svn1024. GNUPG 2.0.16))

I have a number of computer on my home network, running both Linux and Windows. I can create key pairs on each computer, export the public keys and send encrypted files between the computers.

This has meant that I have multiple private/public key pairs all carrying the same information identifying me. It seems that I should be able to use just 1 key pair between all my computers. I can export the public key OK from 1 computer and import it OK on another. But I cannot do the same with the private key - even using the key backup process. The ‘importing’ computer recognises that it being asked to import a private key but refuses to do so.

Is what I am trying to do illogical, or is there some process error I am making?