Missiles don’t just get fired to one position, they can adjust their heading to hit a moving target.

Yes, but homing on a moving target hundreds of miles away is no easy feat. The missile obviously can't do this by itself (like an air-to-air missile). It needs external guidance. There is no radar that has this type of range so you either need a satellite, or an AWACS airplane. In other words, the kill-chain will be quite complex. You can bet there are multiple countermeasures that can disrupt such a kill-chain (shot down the AWACS airplane, blind the satellite with laser, jam the spectrum for the incoming missile, deploy a smoke screen, etc). The simple indication that the "carrier killer" missiles are not such a sure thing is that the Chinese are busy building aircraft carriers themselves. Are they really stupid?

While this is true, there’s limits to the capabilities of many ballistic missiles, some are incapable of changing trajectory. A sufficiently advanced missile should be able to predict course of the carrier and intercept, but I’m sure enemy capability would be taken into account in a high tension scenario.