Rav Shlomo Zalman Auerbach (Meoray Eish and Minchas Shlomo 14) sides that technically as far as the laws of Muktzah are concerned it is permitted. However in a footnote he writes "All what I have written is only a discussion and not to be relied on, for all of Israel refrained all these years from moving an electric light on Shabbos and it appears as uvdin dchol (the work of the weekday) and also we must worry that people may think it is permitted to move for Halacha does not consider it a fire (and may come to use electricity on Shabbos). And although we do not institute our own Gezeira (Rabbinic Prohabition) Leave all of Israel to continue to treat it as if it is Prohibited"
Rav Moshe Feinstein in Igros Moshe (O.C. 5, 32) writes that it is Prohibited since it is like a fire. However he wrote that decision to an english book on the Laws of Muktzah called "Tiltulei Shabbos". Numerous disciples of Rabbi Feinstein testify that he himself carried a flashlight on Shabbos. In sefer "Halacha Shleima" (54:2) writes how he heard from the author of Tiltulei Shabbbos that Rabbi Feinstein held it was permitted and told him to write in english it is assur because it is similar to a fire and people might get confused and come to carry a fire.
There is a specific type of Muktzeh prohibition for a lit fire (See Shabbos 45a). The Chazon Ish gives two reasons to explain why it is Muktzeh. Both of those reasons do not apply to a flashlight and therefore Rabbi Abadi (Ohr Yitzchak O. C. 154) writes that it is permitted. (Rabbi Abadi goes further and writes that "What you heard that Rabbi Feinstein prohibits, maybe Rabbi Feinstein says this based on his opinion of Shabbos clocks that anything that is clear that Chaza"l would prohibit as a Gezeira, is prohibited as if they made such a decree.")
Rav Elyashiv is quoted (Ashrei Ha'ish) to have said since the reasoning of the Muktzeh of a lit fire is not explained in the early Poskim we have no way of knowing a flashlight is not included in the prohibition.
See also Tzitz Eliezer 6 26 who is prohibits it.