The ISIN is the international layer that sits above national schemes. For a US security it is simply the country prefix plus the cusip plus a check digit, so the two identifiers are directly related. Elsewhere the national number comes from the local numbering agency.
ISINs are what cross-border settlement, custody statements, and regulatory reporting run on. They identify the security, not the venue, so one ISIN can trade on several exchanges in several currencies, which is exactly the situation a cross-listing creates.
Example: a US common stock with CUSIP 123456789 has ISIN US1234567897, where US is the country and 7 is the computed check digit. The same ISIN covers the share whether it trades in New York or on a European venue.
Related: cusip, sedol, ticker-symbol, cross-listing, adr