We develop a new method to measure the 3D kinematics of the subphotospheric motion of magnetic elements, which is used to study the coupling between the convection-driven vortex motion and the generation of ubiquitous coronal waves. We use the method of decomposing a line-of-sight magnetogram from MDI/SDO into unipolar magnetic charges, which yields the (projected) 2D motion [x(t), y(t)] and the (half) width evolution w(t) of an emerging magnetic element from an initial depth of d lesssim 1500 km below the photosphere. A simple model of rotational vortex motion with magnetic flux conservation during the emergence process of a magnetic element predicts the width evolution, i.e., w(t)/w 0 = [B(t)/B 0]−1/2, and an upper limit of the depth variation d(t) ≤ 1.3 w(t). While previous 2D tracing of magnetic elements provided information on advection and superdiffusion, our 3D tracing during the emergence process of a magnetic element is consistent with a ballistic trajectory in the upward direction. From the estimated Poynting flux and lifetimes of convective cells, we conclude that the Coronal Multi-channel Polarimeter–detected low-amplitude transverse magnetohydrodynamic waves are generated by the convection-driven vortex motion. Our observational measurements of magnetic elements appear to contradict the theoretical random-walk braiding scenario of Parker.