  We use the Landauer-B\"{u}ttiker formalism to study the mesoscopic Fano
effect in Aharonov-Bohm rings with an embedded two-dimensional noninteracting
dot. The magnetic field dependence of the dot levels leads to a global shift of
the Fano lines which becomes important for small ring/dot area ratios. As the
magnetic field is varied the Fano dips move periodically from one side of the
peak to the other, as reported by Kobayashi {\it et al.} [Phys. Rev. Lett. {\bf
88}, 256806 (2002)]. We show that this effect appears due to a specific
magnetic control of the difference between the phase of the single nonresonant
path via the free arm of the ring and the global phase of all trajectories
involving resonant tunnelings through the dot.
