Fix $k \in \mathbf{Z}$. We first show the map is injective. Suppose that $(\alpha_0, \ldots, \alpha_{r - 1})$ is an element of the left hand side that maps to zero. By Lemma \ref{lemma-cap-projective-bundle} we see that $$ 0 = \pi_*(\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1}) = \alpha_{r - 1} $$ Next, we see that $$ 0 = \pi_*(c_1(\mathcal{O}_P(1)) \cap (\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 2} \cap \pi^*\alpha_{r - 2})) = \alpha_{r - 2} $$ and so on. Hence the map is injective. \medskip\noindent To prove the map is surjective, we will argue exactly as in the proof of Lemma \ref{lemma-pullback-affine-fibres-surjective} to reduce to the case of schemes. We urge the reader to skip the proof. \medskip\noindent Let $\beta \in \CH_{k + r - 1}(P)$. Write $\beta = \sum m_j[W_j]$ with $m_j \not = 0$ and $W_j$ pairwise distinct integral closed subspaces of $\delta$-dimension $k + r$. Then the family $\{W_j\}$ is locally finite in $P$. Let $Z_j \subset X$ be the ``image'' of $W_j$ as in Lemma \ref{lemma-proper-image}. For any quasi-compact open $U \subset X$ we see that $\pi^{-1}(U) \cap W_j$ is nonempty only for finitely many $j$. Hence the collection $Z_j$ of images is a locally finite collection of integral closed subspaces of $X$. \medskip\noindent Consider the fibre product diagrams $$ \xymatrix{ P_j \ar[r] \ar[d]_{\pi_j} & P \ar[d]^\pi \\ Z_j \ar[r] & X } $$ Suppose that $[W_j] \in Z_{k + r - 1}(P_j)$ is rationally equivalent to $$ \pi_j^*\alpha_{j, 0} + c_1(\mathcal{O}(1)) \cap \pi_j^*\alpha_{j, 1} + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi_j^*\alpha_{j, r - 1} $$ for some $(k + i)$-cycle $\alpha_{j, i} \in \CH_{k + i}(Z_j)$. Then $\alpha_i = \sum m_j \beta_{j, i}$ will be a $(k + i)$-cycle on $X$ and $$ \pi^*\alpha_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha_{r - 1} $$ will be rationally equivalent to $\beta$ (see Remark \ref{remark-infinite-sums-rational-equivalences}). This reduces us to the case $X$ integral, and $\alpha = [W]$ for some integral closed subscheme of $P$ dominating $X$. In particular we may assume that $d = \dim_\delta(X) < \infty$. \medskip\noindent Hence we can use induction on $d = \dim_\delta(X)$. If $d < k$, then $\CH_{k + r - 1}(X) = 0$ and the lemma holds; this is the base case of the induction. Consider a nonempty open $U \subset X$. Suppose that we can show that $$ \beta|_{\pi^{-1}(U)} = \pi^*\alpha_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha_{r - 1} $$ for some $\alpha_i \in Z_{k + i}(U)$. By Lemma \ref{lemma-exact-sequence-open} we see that $\alpha_i = \alpha'_i|_U$ for some $\alpha'_i \in Z_{k + i}(X)$. By the exact sequences $\CH_{k + i}(\pi^{-1}(X \setminus U)) \to \CH_{k + i}(P) \to \CH_{k + i}(\pi^{-1}(U))$ of Lemma \ref{lemma-restrict-to-open} we see that $$ \beta - \left(\pi^*\alpha'_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha'_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha'_{r - 1}\right) $$ comes from a cycle $\beta' \in \CH_{k + r}(\pi^{-1}(X \setminus U))$. Since $\dim_\delta(X \setminus U) < d$ we win by induction on $d$. \medskip\noindent In particular, by replacing $X$ by a suitable open we may assume $X$ is a scheme and we have reduced our problem to Chow Homology, Lemma \ref{chow-lemma-chow-ring-projective-bundle}.