* Equities move lower in U.S., Europe
* Dollar drops to 6-1/2 month low versus currency basket
* Tension continues between Russia, Ukraine (Updates to open of U.S. trading)
NEW YORK, May 6 (Reuters) - Stock markets around the world fell on Tuesday as investors found few reasons to keep pushing shares higher, with many indexes near record levels and ongoing geopolitical uncertainty.
The U.S. dollar fell 0.45 percent to a 6-1/2-month low against a basket of currencies, while U.S. bond yields struggled to pull out of their recent troughs and the yen and euro both gained.
U.S. stocks dropped modestly, pressured by financial shares following disappointing results from American International Group. However, the Dow and S&P 500 remained within striking distance of record levels.
Investors continued to watch the ongoing unrest in Ukraine, where more than 30 pro-Russian separatists were killed in fighting near the east Ukraine rebel stronghold of Slaviansk, according to Interior Minister Arsen Avakov. Russia announced plans to beef up its Black Sea warship fleet.
"Valuations are close to the ceiling and some people may be uncomfortable because of that, especially given growing nervousness over the geopolitical tensions," said Bernard Baumohl, managing director and chief global economist at the Economic Outlook Group in Princeton, New Jersey. "However, we still have room to grow before alarm bells start going off."
The Dow Jones industrial average was down 79.98 points, or 0.48 percent, at 16,450.57. The Standard & Poor's 500 Index was down 7.71 points, or 0.41 percent, at 1,876.95. The Nasdaq Composite Index was down 22.14 points, or 0.54 percent, at 4,115.91.
The benchmark 10-year U.S. Treasury note was up 3/32, the yield at 2.5987 percent.
The euro rose 0.4 percent to $1.3931 against the dollar while the yen rose 0.5 percent to 101.50. European shares fell 0.3 percent, while the MSCI International ACWI Price Index slipped 0.1 percent.
Gold dipped 0.1 percent, though prices were supported by the Ukraine tension. Wheat prices - which have been tied to the situation between Russia and Ukraine, both of which are big grain producers- rose 0.6 percent for a third straight daily rise.
In Europe, the European Central Bank is expected to repeat its concern about the strong euro's impact on already-low inflation when it meets on Thursday. But economists doubt the ECB will cut its record-low interest rates again.
Yields on the peripheral countries' lower-rated bonds also remained at multi-year, and in some cases, all-time, lows. Investors welcomed Portugal's plan to exit from its bailout and continued to bet on some future easing of ECB monetary policy. (Reportoing by Ryan vlastelica; Editing by Dan Grebler)