A new resident can be a personal importer of his/her own firearms. Much like swapping driver's licenses, re-registering cars and the like, there are time frames in which it is "required." I put that in quotes because I have little insight into how rigorously some of those things are enforced. Often it comes up, I think, after some sort of problem and one finds themselves on the wrong side of it. Say a car accident, etc. My impression is the DoJ, etc., are more enthusiastic about enforcing firearms things but they have little insight into what you are doing, again, until something happens.
Two things seems to cause confusion. One is the so-called "Handgun Roster." It applies to new sales, FFL transfers and there are limited exceptions. It doesn't apply to private party transfers (the handgun is already legally in the state and transferred between individuals), certain interstate transfers within a limited family relationship. One can personally import handguns which are not on the roster. Incidentally, one can ppt those handguns after they are imported/registered. However one can not import handguns which would be defined under California law as an assault weapon.
One can not import assault weapons (and some other firearms). The Ca. DoJ site goes into detail on what is considered an assault weapon or other prohibitions under Ca. law. Likewise if something is covered by an NFA license, like a suppressor or short barreled something or other, likely a big nope, too. Note that there were a couple of time frames where an "assault weapon" could be registered. Not now. Anything imported must meet current requirements. A possible pitfall? Some of the earliest attempts at defining assault weapons listed some by name. Current requirements are by "feature." One may not apply current legal features to a named AW and import it.
There are stickied threads which would contain the definitions, etc., as well.