It is forbidden on Shabbat to move a candlestick while the candle is burning. And, since the candlestick is forbidden to be moved when Shabbat begins, it remains forbidden to be moved throughout the entirety of Shabbat. There is a Halachic principle of "Migo De’itkasa’i" which establishes that something which is Mukseh during Ben Ha’shemashot (the period after sundown) when Shabbat begins, remains Muskeh the rest of Shabbat. Hence, if a candle was lit in a candlestick before Shabbat, the candlestick may not be moved at any point throughout Shabbat.
There is, however, a way to make it permissible to move the candlestick after the candle is extinguished. Namely, one can stipulate before Shabbat that he wishes to move the candlestick during Shabbat. By making this stipulation, one avoids the provision of "Migo De’itkasa’i" and is then allowed to move the candlestick after the flame goes out. Although the practice among Ashkenazim is not to rely on a stipulation, Sephardic practice allows making this stipulation, in accordance with the ruling of the Hida (Rav Haim Yosef David Auzlai, 1724-1806), in his Birkeh Yosef. The Hida cites his grandfather, Rav Abraham Azulai, as commenting that it suffices to make this stipulation just once a year, and this allows one to move the candlesticks every Shabbat during that year.
Hacham Ovadia Yosef, in Hazon Ovadia (p. 260), writes that once the flame goes out, one who had made the stipulation is then allowed to move the candlesticks for any purpose – whether it’s because he needs them for some permissible purpose, he needs the space, or he wants to protect them from getting ruined. He explains that a candlestick does not fall under the category of "Keli She’melachto Le’issur" – utensils intended mainly for a purpose that is forbidden on Shabbat (such as a hammer), which one may not move on Shabbat for the purpose of protecting them. A candlestick is not used for a forbidden purpose, but rather serves as a "Bassis" (base, or holder) for a candle. As such, although it is Mukseh on Shabbat, it is not considered a "Keli She’melachto Le’issur." Therefore, once a person makes a stipulation, thereby avoiding the prohibition of Mukseh, he is allowed to move the candlestick for any purpose, even to protect it.
For the same reason, Hacham Ovadia adds, candlesticks may be moved even if they are made from silver and are very valuable. The Halacha of "Mukseh Mahamat Hisaron Kis" forbids moving a valuable object which is generally kept in a particular place. However, Hacham Ovadia writes, only something which qualifies as a "Keli She’melachto Le’issur" can be considered "Mukseh Mahamat Hisaron Kis." An expensive painting, for example, is not used for any prohibited purpose, as it just hangs on a wall, and it may therefore be moved on Shabbat despite its being valuable. By the same token, silver candlesticks are not utensils which are used, but rather hold candles. Hence, they do not fall under the category of "Mukseh Mahamat Hisaron Kis," and thus they may be moved on Shabbat after the candles are extinguished (assuming a stipulation was made, as discussed). Likewise, brand new silver candlesticks, which have never been used, may be moved on Shabbat without restriction, for the same reason. (Hacham Ovadia disagrees in this regard with Hacham Bension Abba Shaul, who ruled that valuable candlesticks are considered "Mukseh Mahamat Hisaron Kis" and may not be moved on Shabbat.)
The Shulhan Aruch (Orah Haim 279) cites those who claim that if one places a piece of bread on the tray underneath the candlesticks before Shabbat begins, then the tray with the candlesticks may be moved on Shabbat even if no stipulation was made. Since the tray is now a "base" for both the candlesticks and the bread, it is allowed to be moved. The Shulhan Aruch rejects this opinion, but Hacham Bension Abba Shaul asserted that the Shulhan Aruch referred only to a small piece of bread. If one places a significant piece of bread on the tray, then this would suffice to allow moving the tray with the candlesticks even if no stipulation was made.
Summary: It is forbidden to move candlesticks that were lit when Shabbat began, even after the candles burn out. However, one may make a stipulation before Shabbat that he wishes to move the candlesticks after the flames go out, and then this is allowed. It suffices to make this stipulation once a year, and this stipulation covers every Shabbat throughout the year. If a stipulation is made, one may move the candlesticks after the flames go out for any purpose, even if they are made from silver and are very valuable.