It depends on the system. Some modern systems can react to high-value targets of opportunity, hunt for targets, or switch to a new target if the one they are after is destroyed before they get there. There are different variants of the weapons to deal with different use cases. The 1990s versions were relatively limited though.

Target selection is much more networked, automated, and adaptive than it used to be. Missiles can talk to each other.

Some cruise missiles have the ability to detect targets based on camera or infra-red match; on the other side, most (currently-deployed) drone types have at most that same capability. I believe that most of the infamous Shahed long-range drones that Russia has launched against Ukraine have been entirely inertial or satellite navigation based, with no independent re-targeting capability.

Gulf war tomahawks didn't use GPS. They flew on terrain following radar (over Iran to improve accuracy), inertial reference, and image correlation for the final phase of attack.