Turkey pulled a small military convoy out of a training base in northern Iraq that has become a new flash point as regional powers exert their influence, but refused to buckle to increasing pressure from Baghdad and Moscow to remove all its troops. About 10 Turkish military vehicles left the camp early Monday, Turkish and Iraqi officials said, and they are moving to another base in northern Iraq. While Turkish Prime Minister Ahmet Davutoglu said the move was meant to address Baghdad’s concerns, he added that Ankara had no plans to leave the disputed base. It did little to assuage Iraqi officials who see Ankara’s presence as an illegal incursion. “Redeployment is not withdrawal,” said Iskander Witwit, a member of Iraq’s parliament who sits on the defense and security committee. But U.S. Vice President Joe Biden called Mr. Davutoglu to praise the decision as a positive move to help defuse tensions. “The vice president encouraged the Turkish government to continue its dialogue with Baghdad on additional measures to improve relations between Turkey and Iraq,” the White House said of the call. Turkey prompted a furor two weeks ago when it sent in hundreds of troops and a few dozen tanks to the Bashika base just north of Mosul, which has been controlled by Islamic State since it seized the major city 18 months ago. While Turkish officials initially cast the move as a routine troop rotation that was approved by Iraqi leaders, Baghdad officials demanded that Turkey immediately withdraw. The small base has become just the latest point of conflict for world leaders trying to direct the outcome of the expanding international fight against Islamic State, efforts complicated by conflicting priorities, strategies and regional political rivalries. Moscow—which has seen its own spike in tension with Ankara since a Turkish jet shot down a Russian bomber three weeks ago— has stepped in to support Baghdad in its fight against the Turkish troop moves. Russia and Iraq are calling on the United Nations Security Council to condemn the Turkish troop buildup. Turkey has about 1,000 troops at the disputed base. Turkish leaders said they needed to send in the armored tanks to provide protection for Turkish troops based close to Islamic State strongholds to the south. American officials have urged Turkey to coordinate its military moves with Baghdad, but stopped short of calling for their withdrawal from Iraq. “Turkish soldiers will continue to be there for training,” Mr. Davutoglu said in an interview with A Haber, a privately owned Turkish news channel. “The adjustment today is for the safety of our troops.” Turkey has aligned itself with the U.S.-led coalition battling Islamic State and leaders in northern Iraq’s Kurdish region. Russia is using its bombers to help Syrian President Bashar al-Assad by targeting a variety of Syrian rebels backed by the U.S., Turkey and their allies. Tensions between Ankara and Moscow have simmered. Turkish officials demanded a meeting with Russia ambassador last week after a Russian sailor appeared to brandish a shoulder-fired missile launcher while sailing through Istanbul’s Bosporus Strait. On Sunday, a Russian warship in the Aegean Sea opened fire on a Turkish fishing boat that Moscow said was getting too close, although crew on the fishing boat later told Turkish reporters that they heard no shots and never got close to the Russian ship. Last week, Turkey’s spy chief, Hakan Fidan, and a top Turkish diplomat traveled to Baghdad to try to ease tensions. After the meeting, Turkey said the two countries had agreed on a plan to redeploy Turkish troops from the disputed base, setting the stage for Monday’s move. Turkish troops have been training Kurdish and Sunni forces in northern Iraq for months. Mr. Davutoglu visited some of the training centers this year, and the issue didn’t cause a stir with Baghdad’s Shiite-led government until Turkey sent in the beefed-up force to northern Iraq, where the Kurdish government is locked in a long-standing feud with leaders in the Iraqi capital. Kurdish officials said Masoud Barzani, president of Iraq’s semiautonomous Kurdish region, played a key role in easing tensions by meeting with Turkish leaders last week in Ankara and urging them to work out a solution with Baghdad. The Turkish prime minister said Moscow was only stepping in to condemn Turkey over its troops in Iraq because of their own feud with Ankara. “This act of Russia is not because of their solidarity with the Iraqi government, but because of their reaction to Turkey,” he said last week.