Let $X$ be a compact Hausdorff space. Then if $f: A \to B$ is a map between discrete spaces, the induced map $f^\ast: X^B \to X^A$ is proper.
Question: Are there other classes of map $f: A \to B$ such that $f^\ast: X^B \to X^A$ is proper for every compact Hausdorff space $X$?
Notes:
Part of what I find puzzling about the state of affairs is that when $A$ and $B$ are discrete, we don't need to assume that $f$ itself is proper in order to conclude that $f^\ast$ is proper.
Of course, in order for the question to make sense, we should probably assume that $A$ and $B$ are exponentiable (locally compact Hausdorff is fine). For all I know, though, it may be the case that there are spaces $A$ which are not exponentiable in general, but for which $X^A$ exists for every compact Hausdorff space; the question would make sense for $A,B$ of this form as well.
I'm not certain that restricting $X$ to be compact Hausdorff is the most interesting thing to do, so I'd be interested in this question for other $X$'s as well.
EDIT: I think it's the case that
Claim 0: If $f: A \to B$ is any map between compact Hausdorff spaces, then $f^\ast : X^B \to X^A$ is proper for every compact Hausdorff space $X$.
But I'd like to expand beyond the case where $A,B$ are compact Hausdorff. Moreover, there might be holes in the following argument, as I don't have all this point-set topology at my fingertips:
Claim 1: If $Y,Z$ are compact Hausdorff spaces, then a map $Y \to Z$ is continuous iff it is uniformly continuous (with respect to the unique uniformities consistent with the topologies on $Y,Z$) and moreover, the compact-open topology on $Z^Y$ agrees with the topology of uniform convergence.
Claim 2: If $Y$ is a uniform space and $X$ is compact Hausdorff, then a set $\mathcal C \subseteq X^Y$ (where $X^Y$ is the set of uniformly continuous functions $Y \to X$) is compact in the topology of uniform convergence if and only if it is closed and equicontinuous in the sense that for any $W \subseteq X \times X$ in the uniformity, the set $\{(y,y') \in Y \times Y \mid \forall \varphi \in \mathcal C\, (\varphi(y),\varphi(y')) \in W\}$ is in the uniformity on $Y$.
Claim 3: If $Y,Z,X$ are uniform spaces and $f: Y \to Z$ is uniformly continuous, then $(f^\ast)^{-1}$ preserves equicontinuity (where $f^\ast: X^Z \to X^Y$ is the induced map). That is, if $\mathcal C \subseteq X^Y$ is equicontinuous, then $(f^\ast)^{-1}(\mathcal C) \subseteq X^Z$ is also equicontinuous.
Proof of Claim 0: Let $f: A \to B$ be a map of compact Hausdorff spaces and let $X$ be compact Hausdorff. By Claim 1, we know that $f$ is uniformly continuous and moreover our claim says equivalently that $f^\ast : X^B \to X^A$ is proper with respect to the topology of uniform convergence. By Claim 2, it will suffice to show that $f^{-1}$ preserves equicontinuity, which is the case by Claim 3.