In California, a conservator cannot just move a conservatee out of the state without a court order authorizing the move.
Under Probate Code section 2352, the rules governing a conservatee's change of residence are stated. Probate Code section 2352(c) requires permission of the court to establish a conservatee's residence out of California. A formal petition with notice of hearing is required to all persons who are legally entitled to notice.
Probate Code section 2352(d) states that an order under subdivision (c) requires the conservator to either return the conservatee to California or to cause a conservatorship (or its equivalent) to be commenced in the place of the new residence, when the conservatee has resided in the place of the new residence for a period of four months or a longer/shorter period of time specified in the order.
So in other words, before a conservator decides to move a conservatee out of the state or out of the country, they will need to consult with and comply with Probate Code section 2352 at minimum.