Flying Tiger is correct. Sorry if my original post was unclear. :oops: I’ll try again.
You place an existing fighter on a new carrier by moving the fighter in either combat or noncombat movement, whichever applies, into the sea zone where the new carrier will be mobilized. Air units don’t actually “land” until the end of the Mobilize New Units phase, though they must be in the space in which they will land by the end of noncombat movement. The active player may only move pieces from one space to another during the combat movement and noncombat movement phases.
In Classic A&A, you can’t place fighters (new or existing) on new carriers at all. In Revised, you can place existing fighters on a new carrier in the way you described in your original post. In LHTR, it works the same way as it does in AA50.