Lets say we have a frictionless and massless pulley with a massless string over it. There are two masses attached to it $m_1$ and $m_2$ where $m_2>m_1$.

So $m_2$ would an experience an acceleration down while $m_1$ experiences an acceleration up and massless string would also move accordingly.

However, if the net force on a massless string is always zero, it shouldn't experience any acceleration and stay at rest. But the string does move. How does it move?