First of all, thank you very much for taking the time to post here! :^)
Certificates are linked to your e-mail address and, therefore, should work on multiple machines — although doing so increases the risk that your private key will be compromised.
However, in order to install the certificate on a second computer, you will need to copy the file you downloaded by using Firefox (the one you import in your Keychain) and not something that already is in your Keychain as doing so is likely to leave elements behind. Maybe one of the keys or certificates got lost in the transfer?