The PHB states that when using combat expertise "the changes to attack rolls and Armor Class last until your next action."
So when you attack with combat expertise and then use a move action that means you lose your AC bonus. Is that correct?
When the feat Combat Expertise says, "The changes to attack rolls and Armor Class last until your next action" (Player's Handbook 92), that's legacy text from when the game—largely prior to the 3.5 revision—equated action with turn. That the feat Combat Expertise was never updated to more accurate language—even in the premium edition Player's Handbook (2012)—is a sad editorial oversight.
This issue of action-as-turn persists in pockets throughout 3.5e, but folks tend to ignore the absolutely literal readings in favor of common sense ones, recognizing that the game is, in parts, flawed. For example, a strict reading makes the duration of the effect of the feat Stunning Fist baffling: "A defender who fails this saving throw [against a Stunning Fist attempt] is stunned for 1 round (until just before your next action)" (PH 101), a contradiction that possibly could see a monk stunning his foe and, on the same turn, fleeing from that foe so as to render his foe unstunned! Likewise, making a shield bash would lead to odd battlefield behavior, as making a shield bash usually means "you lose [the shield's] AC bonus until your next action (usually until the next round)" (PH 125)—so a strict reading enables a creature to, for example, make a shield bash then take a free action on its turn to drop prone and regain its shield's bonus to AC! While such readings are possible, such readings are not the norm.
More egregiously, the Player's Handbook Combat chapter is rife with action being used as a synonym for turn. For example, Combat Basics on Initiative says, "A character is flat-footed until he or she takes an action" (135), and taking a free action to speak can typically be done even if it's not a creature's turn! However, I've never heard of a DM saying that a flat-footed creature that yells Hey, rube! off-turn is no longer flat-footed.
In sum, a creature that uses the feat Combat Expertise continues experiencing that feat's bonuses and penalties while it takes the action that yielded the bonuses or penalties and after having taken the action that yielded the bonuses or penalties until the creature's next turn—even if, for example, the creature subsequently takes other actions during this turn or an immediate action or an appropriate free action like speaking after its turn yet before its next.
Note: Many spells that date to before the 3.5 revision have the Casting Time: 1 action entry in their headers—short for 1 standard action. Just one more reason action and turn should never have meant the same thing!
No, it is not.
Unfortunately, early on the rules used this “your action” terminology to refer to your turn. You retain Combat Expertise bonuses until your next turn, e.g. you keep them while moving and then also during other people’s turns. The feat would be even more useless than it already is if it only gave you an AC bonus while attacking (it is very rare to be attacked in the middle of your own attack).