The Shulhan Aruch in Siman 675 establishes an important Halachic principle: “Hadlakah Osah Misva”-The act of lighting the Menorah fulfils the Misva, as opposed to “Hanaha Osah Misva”-the placement of the Menorah is the Misva. That is, it makes no difference if the Menorah was set up by a minor or others who are exempt from the Misva; one would not have to set it up again. This is also true regarding the Shabbat and Yom Tob candles.
There is a discussion amongst the Poskim whether may light the Menorah in one place and then move it to another place. All agree that it is preferable to leave it in the place it was lit. Shulhan Aruch (675:1) clearly writes that the Menorah should not be moved after it was lit, because people will assume that he lit it for his own benefit like a lantern. The Shulhan Aruch also rules that one may not light the Menorah and hold it in his hands for the full half hour. This also appears that he is using it for his own benefit.
Based on this, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) ruled that a sick person who is bedbound upstairs should not light the Menorah in his bedroom and then have it brought downstairs to his window. Rather he should appoint an agent to light for him downstairs.
Hacham Ovadia writes that today the circumstances have changed. It is not such a severe concern that people will think he is using the Menorah for his own benefit, since we use a special Menorah designated for the Misva. Therefore, even if he moved around with it, nobody would assume that he is using it as a lantern. In the olden days, people used regular candles both for the Misva and for illumination, so it was easy to get confused. Therefore, if one moved his Menorah, his Misva is not invalidated. This is also the opinion of the Mishna Berura in Siman 675. In the case of the sick person, Hacham Ovadia would allow him to light in his bedroom and then have it brought downstairs, but he still prefers the option of appointing an agent.
It is also preferable not to move the Shabbat and Yom Tob candles after they have been lit. (Although not preferable, one may move Shabbat candles as long as hee didn’t accept Shabbat yet.) Although the Taz (Rabbi David Segal, Poland, 1586-1667) holds that Shabbat and Yom Tob are different, The Hida (Rav Haim Yosef David Azulai, 1724-1806) defends the position of those who equate the two Misvot. The way to remember this Halacha is that the Beracha for all these Misvot is “L’Hadlik”-to light-meaning that the lighting is the Misva.
It is preferable not to move the Chanukah candles after they have been lit, but doing do would not invalidate the Misva..