By the result of Lemma \ref{lemma-compose-base-change} we obtain an inverse system of chow groups $\CH_k(X_i)$ and a map $\colim \CH_i(X_i) \to \CH_k(X)$. We may replace $S$ by a quasi-compact open through which $X \to S$ factors, hence we may and do assume all the schemes occuring in this proof are Noetherian (and hence quasi-compact and quasi-separated). \medskip\noindent Let us show that this map is surjective. Namely, let $Z' \subset X'$ be an integral closed subscheme of $\delta'$-dimension $k$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an $i$ and a morphism $Z_i \to X_i$ of finite presentation whose base change is $Z'$. Afer increasing $i$ we may assume $Z_i$ is a closed subscheme of $X_i$, see Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation}. Then $Z' \to X_i$ factors through $Z_i$ and we may replace $Z_i$ by the scheme theoretic image of $Z' \to X_i$. In this way we see that we may assume $Z_i$ is an integral closed subscheme of $X_i$. By Lemma \ref{lemma-dimension-base-change} we conclude that $\dim_{\delta_i}(Z_i) = \dim_{\delta'}(Z') = k$. Thus $\CH_k(X_i) \to \CH_k(X')$ maps $[Z_i]$ to $[Z']$ and we conclude surjectivity holds. \medskip\noindent Let us show that our map is injective. Let $\alpha_i \in \CH_k(X_i)$ be a cycle whose image $\alpha' \in \CH_k(X')$ is zero. Then there exist integral closed subschemes $W_l' \subset X'$, $l = 1, \ldots, r$ of $\delta"$-dimension $k + 1$ and nonzero rational functions $f'_l$ on $W'_l$ such that $\alpha' = \sum_{l = 1, \ldots, r} \text{div}_{W'_l}(f'_l)$. Arguing as above we can find an $i$ and integral closed subschemes $W_{i, l} \subset X_i$ of $\delta_i$-dimension $k + 1$ whose base change is $W'_l$. After increasin $i$ we may assume we have rational functions $f_{i, l}$ on $W_{i, l}$. Namely, we may think of $f'_l$ as a section of the structure sheaf over a nonempty open $U'_l \subset W'_l$, we can descend these opens by Limits, Lemma \ref{limits-lemma-descend-opens} and after increasing $i$ we may descend $f'_l$ by Limits, Lemma \ref{limits-lemma-descend-section}. We claim that $$ \alpha_i = \sum\nolimits_{l = 1, \ldots, r} \text{div}_{W_{i, l}}(f_{i, l}) $$ after possibly increasing $i$. \medskip\noindent To prove the claim, let $Z'_{l, j} \subset W'_l$ be a finite collection of integral closed subschemes of $\delta'$-dimension $k$ such that $f'_l$ is an invertible regular function outside $\bigcup_j Y'_{l, j}$. After increasing $i$ (by the arguments above) we may assume there exist integral closed subschemes $Z_{i, l, j} \subset W_i$ of $\delta_i$-dimension $k$ such that $f_{i, l}$ is an invertible regular function outside $\bigcup_j Z_{i, l, j}$. Then we may write $$ \text{div}_{W'_l}(f'_l) = \sum n_{l, j} [Z'_{l, j}] $$ and $$ \text{div}_{W_{i, l}}(f_{i, l}) = \sum n_{i, l, j} [Z_{i, l, j}] $$ To prove the claim it suffices to show that $n_{l, i} = n_{i, l, j}$. Namely, this will imply that $\beta_i = \alpha_i - \sum\nolimits_{l = 1, \ldots, r} \text{div}_{W_{i, l}}(f_{i, l})$ is a cycle on $X_i$ whose pullback to $X'$ is zero as a cycle! It follows that $\beta_i$ pulls back to zero as a cycle on $X_{i'}$ for some $i' \geq i$ by an easy argument we omit. \medskip\noindent To prove the equality $n_{l, i} = n_{i, l, j}$ we choose a generic point $\xi' \in Z'_{l, j}$ and we denote $\xi \in Z_{i, l, j}$ the image which is a generic point also. Then the local ring map $$ \mathcal{O}_{W_{i, l}, \xi} \longrightarrow \mathcal{O}_{W'_l, \xi'} $$ is flat as $W'_l \to W_{i, l}$ is the base change of the flat morphism $S' \to S_i$. We also have $\mathfrak m_\xi \mathcal{O}_{W'_l, \xi'} = \mathfrak m_{\xi'}$ because $Z_{i, l, j}$ pulls back to $Z'_{l, j}$! Thus the equality of $$ n_{l, j} = \text{ord}_{Z'_{l, j}}(f'_l) = \text{ord}_{\mathcal{O}_{W'_l, \xi'}}(f'_l) \quad\text{and}\quad n_{i, l, j} = \text{ord}_{Z_{i, l, j}}(f_{i, l}) = \text{ord}_{\mathcal{O}_{W_{i, l}, \xi}}(f_{i, l}) $$ follows from Algebra, Lemma \ref{algebra-lemma-pullback-module} and the construction of $\text{ord}$ in Algebra, Section \ref{algebra-section-orders-of-vanishing}.