Since every power object is an internal Heyting algebra, and $f_*$ preserves the structure of internal Heyting algebras, there are trivial examples of such natural transformations corresponding to the constants $\top$ and $\bot$. Of course, this is uninteresting.
Let me write $P^{\mathcal{D}}$ and $P^{\mathcal{E}}$ for the respective contravariant power object functors. Since $f_*$ preserves monomorphisms, there is a canonical comparison morphism $f_* \Omega_{\mathcal{D}} \to \Omega_{\mathcal{E}}$; since $f_*$ preserves products, there is a canonical natural morphism $f_* (Y^X) \to (f_* Y)^{f_* X}$; and so there is a canonical natural morphism $f_* P^{\mathcal{D}} X \to (f_* \Omega_{\mathcal{D}})^{f_* X} \to P^{\mathcal{E}} f_* X$. So there is an interesting canonical natural transformation $\theta : f_* P^{\mathcal{D}} \Rightarrow P^{\mathcal{E}} f_*$.
Now allow me to argue using generalised elements. Let $T$ be an arbitrary object of $\mathcal{E}$, and let $p : X \to Y$ be a morphism in $\mathcal{D}$. Given a generalised element $t : T \to f_* P_\mathcal{D} X$, what is $\theta_X \circ t : T \to P_{\mathcal{E}} f_{\ast} X$, and what is $f_* \exists_p \circ t : T \to f_* P_{\mathcal{D}} Y$? Let $t' : f^* T \to P_\mathcal{D} X$ be the left adjoint transpose of $t$, and let $A' \rightarrowtail X \times f^* T$ be the subobject classified by $t'$. It is clear that $\theta_X \circ t$ is just the classifying morphism for the pullback of $f_* A' \rightarrowtail f_* X \times f_* f^* T$ along $f_* X \times T \to f_* X \times f_* f^* T$. Also, by naturality, $f_* \exists_p \circ t$ must be the right adjoint transpose of $\exists_p \circ t' : f^* T \to P_\mathcal{D} Y$, which is none other than the classifying morphism for the image of the composite $A' \rightarrowtail X \times f^* T \to Y \times f^*T$.
This suggests the crucial criterion is that $f_*$ preserve epimorphisms (and hence, epi–mono factorisations) – and this automatic for all base change morphisms for slices over $\textbf{Set}$ because $\textbf{Set}$ and its slices have the axiom of choice. So assume $f_*$ preserves epimorphisms. If we write $A \rightarrowtail f_* X \times T$ for the subobject classified by $\theta_X \circ t$, $B' \rightarrowtail Y \times f^* T$ for the image of $A' \rightarrowtail X \times f^* T \to Y \times f^* T$, and $B \rightarrowtail f_* Y \times T$ for the subobject classified by $\theta_Y \circ f_* \exists_p \circ t$, then the preservation of epi–mono factorisations implies that $f_* B'$ remains the image of $f_* A'$ under $f_* X \times f_* f^* T \to f_* Y \times f_* f^* T\ $; but epi–mono factorisations are stable under pullback in a topos, hence $B$ is the image of $A$ under $f_* X \times T \to f_* Y \times T$. Thus, we have $$\theta_Y \circ f_* \exists_p \circ t = \exists_{f_{\ast} p} \circ \theta_X \circ t$$ for all generalised elements $t : T \to f_* P_{\mathcal{D}} X$, and thus $\theta$ is also a natural transformation $f_* P_{\mathcal{D}} \Rightarrow f_* P_{\mathcal{E}}$.
