We first remark that $X$ is a locally Noetherian algebraic space by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. Hence the statement of the lemma makes sense. Moreover, computing $R^if_*\mathcal{F}$ commutes with \'etale localization on $Y$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}) and checking whether $R^if_*\mathcal{F}$ coherent can be done \'etale locally on $Y$ (Lemma \ref{lemma-coherent-Noetherian}). Hence we may assume that $Y = \Spec(A)$ is a Noetherian affine scheme. \medskip\noindent Assume $Y = \Spec(A)$ is an affine scheme. Note that $f$ is locally of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}). Thus it is of finite presentation, hence $X$ is Noetherian (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}). Thus Lemma \ref{lemma-property-higher-rank-cohomological-variant} applies to the category of coherent modules of $X$. For a coherent sheaf $\mathcal{F}$ on $X$ we say $\mathcal{P}$ holds if and only if $R^if_*\mathcal{F}$ is a coherent module on $\Spec(A)$. We will show that conditions (1), (2), and (3) of Lemma \ref{lemma-property-higher-rank-cohomological-variant} hold for this property thereby finishing the proof of the lemma. \medskip\noindent Verification of condition (1). Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of higher direct images $$ R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1 $$ Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$ have property $\mathcal{P}$, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemmas \ref{lemma-coherent-abelian-Noetherian} and \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. Hence property $\mathcal{P}$ holds for the third as well. \medskip\noindent Verification of condition (2). This follows immediately from the fact that $R^if_*(\mathcal{F}_1 \oplus \mathcal{F}_2) = R^if_*\mathcal{F}_1 \oplus R^if_*\mathcal{F}_2$ and that a summand of a coherent module is coherent (see lemmas cited above). \medskip\noindent Verification of condition (3). Let $i : Z \to X$ be a closed immersion with $Z$ reduced and $|Z|$ irreducible. Set $g = f \circ i : Z \to \Spec(A)$. Let $\mathcal{G}$ be a coherent module on $Z$ whose scheme theoretic support is equal to $Z$ such that $R^pg_*\mathcal{G}$ is coherent for all $p$. Then $\mathcal{F} = i_*\mathcal{G}$ is a coherent module on $X$ whose support scheme theoretic support is $Z$ such that $R^pf_*\mathcal{F} = R^pg_*\mathcal{G}$. To see this use the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}) and the fact that $R^qi_*\mathcal{G} = 0$ for $q > 0$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images} and the fact that a closed immersion is affine. (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-affine}). Thus we reduce to finding a coherent sheaf $\mathcal{G}$ on $Z$ with support equal to $Z$ such that $R^pg_*\mathcal{G}$ is coherent for all $p$. \medskip\noindent We apply Lemma \ref{lemma-weak-chow} to the morphism $Z \to \Spec(A)$. Thus we get a diagram $$ \xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_A \ar[dl] \\ & \Spec(A) & } $$ with $\pi : Z' \to Z$ proper surjective and $i$ an immersion. Since $Z \to \Spec(A)$ is proper we conclude that $g'$ is proper (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-proper}). Hence $i$ is a closed immersion (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-universally-closed-permanence} and \ref{spaces-morphisms-lemma-immersion-when-closed}). It follows that the morphism $i' = (i, \pi) : \mathbf{P}^n_A \times_{\Spec(A)} Z' = \mathbf{P}^n_Z$ is a closed immersion (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-semi-diagonal}). Set $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_A}(1) = (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1) $$ We may apply Lemma \ref{lemma-kill-by-twisting} to $\mathcal{L}$ and $\pi$ as well as $\mathcal{L}$ and $g'$.  Hence for all $d \gg 0$ we have $R^p\pi_*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$ and $R^p(g')_*\mathcal{L}^{\otimes d} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes d}$. By the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}) we have $$ E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes d} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes d} $$ and by choice of $d$ the only nonzero terms in $E_2^{p, q}$ are those with $q = 0$ and the only nonzero terms of $R^{p + q}(g')_*\mathcal{L}^{\otimes d}$ are those with $p = q = 0$. This implies that $R^pg_*\mathcal{G} = 0$ for $p > 0$ and that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$. Applying Cohomology of Schemes, Lemma \ref{coherent-lemma-locally-projective-pushforward} we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes d}$ is coherent. \medskip\noindent We still have to check that the support of $\mathcal{G}$ is $Z$. This follows from the fact that $\mathcal{L}^{\otimes d}$ has lots of global sections. We spell it out here. Note that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq 0$ because the same is true for $\mathcal{O}_{\mathbf{P}^n}(d)$. Pick a point $z \in Z'$ mapping to the generic point $\xi$ of $Z$ which we can do as $\pi$ is surjective. (Observe that $Z$ does indeed have a generic point as $|Z|$ is irreducible and $Z$ is Noetherian, hence quasi-separated, hence $|Z|$ is a sober topological space by Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-separated-sober}.) Pick $s \in \Gamma(Z', \mathcal{L}^{\otimes d})$ which does not vanish at $z$. Since $\Gamma(Z, \mathcal{G}) = \Gamma(Z', \mathcal{L}^{\otimes d})$ we may think of $s$ as a global section of $\mathcal{G}$. Choose a geometric point $\overline{z}$ of $Z'$ lying over $z$ and denote $\overline{\xi} = g' \circ \overline{z}$ the corresponding geometric point of $Z$. The adjunction map $$ (g')^*\mathcal{G} = (g')^*g'_*\mathcal{L}^{\otimes d} \longrightarrow \mathcal{L}^{\otimes d} $$ induces a map of stalks $\mathcal{G}_{\overline{\xi}} \to \mathcal{L}_{\overline{z}}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback-quasi-coherent}. Moreover the adjunction map sends the pullback of $s$ (viewed as a section of $\mathcal{G}$) to $s$ (viewed as a section of $\mathcal{L}^{\otimes d}$). Thus the image of $s$ in the vector space which is the source of the arrow $$ \mathcal{G}_{\overline{\xi}} \otimes \kappa(\overline{\xi}) \longrightarrow \mathcal{L}^{\otimes d}_{\overline{z}} \otimes \kappa(\overline{z}) $$ isn't zero since by choice of $s$ the image in the target of the arrow is nonzero. Hence $\xi$ is in the support of $\mathcal{G}$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}). Since $|Z|$ is irreducible and $Z$ is reduced we conclude that the scheme theoretic support of $\mathcal{G}$ is all of $Z$ as desired.