Optical patterns with the orbital angular momentum (OAM) can serve as “optical spanners”, and have attracted much attention for their applications in biophysics, micromechanics or microfluidics. In this work, we theoretically demonstrate the propagation characteristics of elliptic vortex beams carrying the OAM in nonlocal nonlinear media. The elliptic vortex beams cannot evolve as solitons due to their non-circular-symmetry, but can evolve as breathers in the nonlinear media without anisotropy. During breathing evolutions, the rotating phenomenon is observed. The rotations take place in both the vortex cores and the beam envelopes, which exhibit opposite rotating directions. The breathing and rotating evolutions both have close relations with the input power and the beam size. The analytical results are well confirmed by the numerical simulations. The results obtained in the paper might find potential applications in beam shaping, and in optical manipulations of rod-shaped micro-particles.