The assumptions are preserved by base change with a morphism $X' \to X$ locally of finite type. Hence it suffices to show that $P_p(Z \to X, F) \cap \alpha = P'_p(Q) \cap \alpha$, resp.\ $c_p(Z \to X, F) \cap \alpha = c'_p(Q) \cap \alpha$ for any $\alpha \in \CH_k(X)$. Choose $\beta \in \CH_{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_X)$ is equal to the flat pullback of $\alpha$ as in the construction of $C$ in Lemma \ref{lemma-gysin-at-infty}. Denote $E \subset W_\infty$ the inverse image of $Z$. \medskip\noindent Let $U \subset X$ be the maximal open subscheme such that $F|_U$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_U$-module of rank $< p$ sitting in cohomological degree $0$. Let $V \subset W$ be the maximal open subscheme such that $Q|_V$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_V$-module of rank $< p$ sitting in cohomological degree $0$. By our assumptions on $Q$ and $F$ we have $$ V \cap b^{-1}(\mathbf{A}^1_X) = \mathbf{A}^1_U,\quad V \cap W_\infty \supset T, \quad\text{and}\quad X \setminus U \subset Z $$ Let $Z' = X \setminus U$ and let $W' \subset W$ be the scheme theoretic closure of $b^{-1}(\mathbf{A}^1_{Z'})$. The inclusions above imply that we have $Z' \subset Z$, $b(W') \subset \mathbf{P}^1_{Z'}$, $b' : W' \to \mathbf{P}^1_{Z'}$ is an isomorphism over $\mathbf{A}^1_{Z'}$, and that $Q|_{W \setminus W'}$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{W \setminus W'}$-module of rank $< p$ sitting in cohomological degree $0$. The lemma follows from the following sequence of equalities (the case of $P_p$ is similar) \begin{align*} c'_p(Q) \cap \alpha & = (E \to Z)_*(c'_p(Q|_E) \cap i_\infty^*\beta) \\ & = (E \to Z)_*(c_p(E \to W_\infty, Q|_{W_\infty}) \cap i_\infty^*\beta) \\ & = (E \to Z)_*(W'_\infty \to E)_*(c_p(W' \to W, Q) \cap i_\infty^*\beta) \\ & = (W'_\infty \to Z)_*((i'_\infty)^*(c_p(W' \to W, Q) \cap \beta)) \\ & = (W'_\infty \to Z)_*(C' \cap c_p(Z' \to X, F) \cap \alpha) \\ & = c_p(Z \to X, F) \cap \alpha \end{align*} The first equality is the construction of $c'_p(Q)$.  The second is Lemma \ref{lemma-loc-chern-agree}. The third expresses the fact that the restriction of $c_p(W' \to W, Q)$ to $W_\infty$ is equal to $c_p(W'_\infty \to W_\infty, Q|_{W_\infty})$, see for example Lemma \ref{lemma-base-change-loc-chern}, and that $c_p(W'_\infty \to W_\infty, Q|_{W_\infty})$ pushes forward to $c_p(E \to W_\infty, Q|_{W_\infty})$ by Lemma \ref{lemma-loc-chern-shrink-Z}. The fourth is commutation of a bivariant class with a gysin homomorphism. For the fith, please observe that $c_p(W' \to W, Q)$ and $c_p(Z' \to X, E)$ restrict to the same bivariant class on $b^{-1}(\mathbf{A}^1_X)$ by assumption (1) of the lemma. Hence $c_p(W' \to W, Q) \cap \beta$ is a class in $\CH_{k + 1 - p}(W')$ whose restriction to $(b')^{-1}(\mathbf{A}^1_{Z'})$ agrees with the flat pullback of $c_p(Z' \to X, F) \cap \alpha$. Thus $(i'_\infty)^*(c_p(W' \to W, Q) \cap \beta)$ is equal to $C' \cap c_p(Z' \to X, F) \cap \alpha$ where $C' \in A^0(W'_\infty \to Z')$ is constructed in Lemma \ref{lemma-gysin-at-infty}. The final equality holds because $c_p(Z' \to X, F)$ pushes forward to $c_p(Z \to X, F)$ and because $(W'_\infty \to Z)_* \circ C' = 1$ in $A^0(Z)$. In fact we have $(W'_\infty \to Z') \circ C' = 1$ in $A^0(Z')$ as follows from the final statement in Lemma \ref{lemma-gysin-at-infty}.