Consider a family of elliptic curves over the open disc $D$ in $\mathbb{C}$, which degenerate to the nodal elliptic curve over $0$, and let $f$ be the map to $D$. I'm interested in the sheaf on the central fibre which is the derived pushforward of the constant sheaf on one of the generic fibres, i.e. $Rr_*\mathbb{Q}_{f^{-1}(\epsilon)}$, where $r$ is the quotient map $f^{-1}(\epsilon)\to f^{-1}(0)$.
My question is about why I should expect $Rr_*\mathbb{Q}_{f^{-1}(\epsilon)} = \Psi_f\mathbb{Q}_{f^{-1}(0)}$, where $\Psi_f$ is the nearby cycle functor. Certainly this seems like a nice candidate for the nearby cycle, and in the notes of de Cataldo and Migliorini (https://arxiv.org/pdf/0712.0349.pdf) they state such a property for proper maps $f$ on page 92, in Remark 5.5.1 of the Appendix. They point to section II.6.13 in "Stratified Morse Theory" by Goresky and MacPherson for details, but I am unable to interpret these results in the appropriate way.
Thanks in advance