Optical vortices are ubiquitous in coherent optical beams -- they appear naturally in speckle fields and can also be excited artificially with, for instance, diffractive optical elements. An understanding of the propagation of such vortices would be useful for applications of this phenomenon. A method is provided to compute the trajectories of optical vortices in complicated scenarios. The Gaussian beam in which the vortices are embedded is expressed in terms of paraxial modes. The positions of the vortices as a function of the propagation distance can then be computed analytically. The case of a vortex dipole is analyzed and shown to undergo annihilation and revival of the pair under certain conditions. Expressions are provided for the trajectories of a canonically launched vortex dipole. The analytical predictions are compared with numerical simulations.