Unipotent nearby cycles and nearby cycles over general bases

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $\mathrm{\Psi }$-factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.