CUSIPs, ISINs, SEDOLs, etc are identifiers for securities (stocks, bonds, and sometimes other instruments), not for companies. For example, one company might have several classes of common stock; each has its own cusip. Companies have other identifiers. As Kurtosis wrote, there's an industry trend to use LEIs as company id's.
Empirically, the various organizations that assign identifiers to securities usually don't change bond identifiers, but do change stock identifiers when a company/issuer merely changes name or reorganizes.
I'm not aware of any written rules, but most corporate actions seem to result in a "new" stock with new identifiers. (But be careful, occasionally a merger or a spin-off totally changes the nature of the company, yet the stock identifiers don't change.)
Typical scenarios where bond identifier changes might be: a company issues 144A and Reg S tranches of a bond, each with its own ISIN. Some time later the company issues a Global tranche, with a new ISIN, and the old tranches are converted to the new one and cease to exist. A company issues a bond, goes through bankruptcy, issues a totally new bond with longer maturity given to old bondholders in exchange for the old one.
But most corporate actions don't change a bond id, even through the bond might have a totally different issuer/guarantor than when it weas issued.
CRSP database has a good history of corporate actions going back decades, but is very U.S.-centric.
