We study the propagation of off-axis vortices in a paraxial beam formed by two collinear Laguerre-Gauss beams. We show that the vortices move about the beam axis as the light propagates resulting in a rotation of the beam's transverse profile. This rotation is explained by the Gouy phase acquired by the component beams. Experimental measurements of the angular position of the vortices are in good agreement with a two-mode theory.