The dynamic response of Barium Strontium Titanate (Ba1-xSrxTiO3, x=0.6-0.9) is presented over a temperature range spanning various structural phase transitions and for compositions that extend from the ferroelectric (FE) phase to the FE-Antiferrodistortive (AFD) phase boundary. We find that while the in-phase part of the dielectric response is insensitive to applied frequencies over the range 400-500kHz for all compositions, strong dispersive effects are observed in general for the out of phase part, for almost all compositions. The dielectric peaks corresponding to the lower temperature structural transitions (e.g. tetragonal to orthorhombic T-O etc.) exhibit frequency dependence, while the higher temperature cubic to tetragonal (C-T) transition does not show dispersive effects. The only exception to this is for the Ba0.1Sr0.9TiO3 composition, lying at the boundary of the FE and AFD phases. Detailed frequency dependence studies of this composition (x=0.9) show that the (C-T) transition exhibits an abrupt change in the dispersive features for high frequencies f>10kHz. These features include the constancy of peak temperature, large drop in the magnitude of the losses and a noticeable broadening of the associated loss peak. The anomalous features in the behavior of x=0.9 composition are attributed to slow relaxation processes in the AFD phase at the boundary separating the FE and AFD phases.