Part (1) follows from (2) and Lemma \ref{lemma-cycle-closed-coherent}. Let $\mathcal{F}$ be a coherent sheaf on $X$. Assume that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-coherent-support-closed} there exists a closed immersion $i : Z \to X$ and a coherent $\mathcal{O}_Z$-module $\mathcal{G}$ such that $i_*\mathcal{G} \cong \mathcal{F}$ and such that the support of $\mathcal{F}$ is $Z$. Let $Z' \subset Y$ be the scheme theoretic image of $f|_Z : Z \to Y$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-image}. Consider the commutative diagram $$ \xymatrix{ Z \ar[r]_i \ar[d]_{f|_Z} & X \ar[d]^f \\ Z' \ar[r]^{i'} & Y } $$ of algebraic spaces over $B$. Observe that $f|_Z$ is surjective (follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image} and the fact that $|f|$ is closed) and proper (follows from Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-proper}, \ref{spaces-morphisms-lemma-closed-immersion-proper}, and \ref{spaces-morphisms-lemma-universally-closed-permanence}). We have $f_*\mathcal{F} = f_*i_*\mathcal{G} = i'_*(f|_Z)_*\mathcal{G}$ by going around the diagram in two ways. Suppose we know the result holds for closed immersions and for $f|_Z$. Then we see that $$ f_*[\mathcal{F}]_k = f_*i_*[\mathcal{G}]_k = (i')_*(f|_Z)_*[\mathcal{G}]_k = (i')_*[(f|_Z)_*\mathcal{G}]_k = [(i')_*(f|_Z)_*\mathcal{G}]_k = [f_*\mathcal{F}]_k $$ as desired. The case of a closed immersion follows from Lemma \ref{lemma-length-closed-immersion} and the definitions. Thus we have reduced to the case where $\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and surjective. \medskip\noindent Assume $\dim_\delta(X) \leq k$ and $f : X \to Y$ is proper and surjective. For every irreducible component $Z \subset Y$ with generic point $\eta$ there exists a point $\xi \in X$ such that $f(\xi) = \eta$. Hence $\delta(\eta) \leq \delta(\xi) \leq k$. Thus we see that in the expressions $$ f_*[\mathcal{F}]_k = \sum n_Z[Z], \quad \text{and} \quad [f_*\mathcal{F}]_k = \sum m_Z[Z]. $$ whenever $n_Z \not = 0$, or $m_Z \not = 0$ the integral closed subspace $Z$ is actually an irreducible component of $Y$ of $\delta$-dimension $k$ (see Lemma \ref{lemma-point-of-max-dimension}). Pick such an integral closed subspace $Z \subset Y$ and denote $\eta$ its generic point. Note that for any $\xi \in X$ with $f(\xi) = \eta$ we have $\delta(\xi) \geq k$ and hence $\xi$ is a generic point of an irreducible component of $X$ of $\delta$-dimension $k$ as well (see Lemma \ref{lemma-point-of-max-dimension}). By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-finite-in-codim-1} there exists an open subspace $\eta \in V \subset Y$ such that $f^{-1}(V) \to V$ is finite. Since $\eta$ is a generic point of an irreducible component of $|Y|$ we may assume $V$ is an affine scheme, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme}. Replacing $Y$ by $V$ and $X$ by $f^{-1}(V)$ we reduce to the case where $Y$ is affine, and $f$ is finite. In particular $X$ and $Y$ are schemes and we reduce to the corresponding result for schemes, see Chow Homology, Lemma \ref{chow-lemma-cycle-push-sheaf} (applied with $S = Y$).