The magnitude and phase of the transmission amplitude of a multilevel quantum dot is calculated for the mesoscopic regime of level spacing large compared to level width. The interplay between Kondo correlations and the influence by neighboring levels is discussed. As in the single-level case, the Kondo plateaus of magnitude and phase disappear with increasing temperature. At certain gate voltages, ``stationary'' points are found at which the transmission phase is independent of temperature. Depending on the mesoscopic parameters of the adjacent levels (such as relative sign and magnitude of tunneling matrix elements), the stationary points are shifted to or repelled by the neighboring level.