WASHINGTON—The U.S. military has begun moving equipment out of Syria as it pushes forward on President Trump’s order to withdraw forces, but planners are still working out details of how and when more than 2,000 troops will leave, defense officials said Friday.

Some logistical and support troops have now moved into Syria, after first going to Kuwait and Iraq, to help with an eventual U.S. withdrawal, defense officials said. At the same time, a small number of U.S. troops operating in Syria left in the past week, and a limited...