From PKI - CA Cross-Certify Certificate with OpenSSL?:
First, use 'openssl x509 -x509toreq' to create a CSR from the certificate you want to cross-certify, then use 'openssl ca' to sign it, and use a specific extension section in the configuration file (use the -extensions option). Alternatively, the CA you want to cross-certify with could send you the CSR they used to create their CA certificate, and you could send them yours.
Cross-certification is really not that magic. You do need to keep track of policies, how you map policies, path lengths and things like that, which is why I suggest you have a specific configuration section for this purpose. Of course, you might just not care, and in that case, the extension v3_ca should be good enough.