In many homes, the Shabbat candles are placed on the Shabbat table, on top of a tray. Clearly, it is forbidden on Shabbat to move candlesticks that were lit before Shabbat and were lit through the onset of Shabbat. The more complicated question is whether one may lift the tray with the candlesticks on them and move them off the table on Shabbat.
The law of "Bassis" ("base") establishes that just as a Mukseh item may not be moved on Shabbat, the utensil upon which it was placed before Shabbat may likewise not be moved on Shabbat. Even though the "Bassis" is intrinsically permissible, and, in and of itself, it may be moved on Shabbat, nevertheless, since a Mukseh item was placed upon it when Shabbat began, it may not be moved on Shabbat. Therefore, as a general rule, it is forbidden to move on Shabbat not only the Shabbat candlesticks, but also the tray upon which they were placed when Shabbat begun.
If one wishes to move the tray and the candlesticks during Shabbat, there are two ways in which this may be done. First, one may ask a non-Jew to move them. Although it is generally forbidden to ask a non-Jew to perform an act on Shabbat which is forbidden for Jews, this does not apply to the Mukseh prohibitions. Therefore, one may ask a non-Jew (such as a non-Jewish housekeeper) on Shabbat to move the candlesticks and the tray off the table. This is the ruling of the Aruch Ha’shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908). It must be emphasized, however, that if a non-Jew moved the candlesticks, it is still forbidden to then remove the tray, as it is considered Mukseh throughout the entirety of Shabbat, even after the candlesticks are removed.
The other option is to place something else significant together with the candlesticks on the tray. If, for example, one places a piece of bread on the tray alongside the candlesticks, the tray is regarded as a "Bassis" for both something forbidden (the candlesticks) and something permissible (the bread). Such a "Bassis" may be moved on Shabbat. Therefore, one who anticipates having to move the candlesticks from the table during Shabbat should make a point of placing something else on the tray before Shabbat, and the tray may then be moved on Shabbat.
The Halachic authorities debate the question of whether under such circumstances one may move the tray while the candles are still burning. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) rules that as long as one carries the tray slowly and carefully, so that the oil in the cups are not jostled, he may move the tray even while the candles burn. Hacham Bension Abba Shaul (Israel, 1923-1998), however, rules that if oil candles are burning, they may not be carried on Shabbat even by carrying the tray. Moving oil candles has the effect of moving the oil away from or closer to the wick, which could transgress the prohibitions of Mab’ir (kindling) and Mechabeh (extinguishing). And even if wax candles are used, Hacham Bension writes, sometimes liquid melted wax collects on the top of the candle, and when this liquid moves it could increase or decrease the flame, in violation of Shabbat. Although the Shulhan Aruch explicitly permits moving wax candles on Shabbat in this fashion, Hacham Bension claims that this refers only to narrow candles (like the candles many people use on Hanukah), on which melted wax does not collect. But the thick wax candles, like those which many women use for Shabbat candles, may not be moved while they are burning, according to his opinion.
Summary: If the Shabbat candlesticks are placed on a tray, the tray may not be moved on Shabbat, even after the candles are no longer burning. One may, however, ask a non-Jew to move the tray and the candlesticks. If one places a piece of food or other significant item on the tray alongside the candlesticks before Shabbat begins, then he may move the tray on Shabbat, though according to some opinions, even then he may not move the tray while the candles burn.