Transverse free vibrations of axially moving nanobeams subjected to axial tension are studied based on nonlocal stress elasticity theory. A lower and higher-order differential equation of motion is derived from the D'Alembert principle with corresponding lower-order and higher-order, non-classical boundary conditions. Effects of nonlocal nanoscale, dimensionless axial velocity, density and axial tension on natural frequencies are presented and discussed through numerical examples for both lower and higher order differential equation of transverse motion. It is found that these factors have great influence on the dynamic behaviour of an axially moving nanobeam. In particular, nonlocal effect "softens" the system, in lower order case and the nonlocal effect "harden" the system, in higher order case. And also found that higher order differential equation of transverse motion with higher non-classical boundary conditions are more effective where increasing nonlocal stress effects increasing natural frequency in fact induce increased nanostructural stiffness,i.e.,decreasing deflection.