Let $X$ and $Y$ be two integral separated Noetherian Gorenstein schemes over a base field $k$ of arbitrary characteristic whose local rings are unique factorization domains and $f: X\to Y$ an étale map between them. Since we assumed these to be Gorenstein, their canonical bundles $\omega_X$, resp. $\omega_Y$ exist and are line bundles / invertible sheaves.

The map $f$ induces via pullback the map $f^*: \text{Pic}(Y) \to \text{Pic}(X)$ between Picard groups and the rather natural question which arrises at this point is how $\omega_X$ and $f^*\omega_Y$ are related to each other?

Since $X$ and $Y$ where moreover assumed to be integral separated and locally factorial and therefore the Cartier divisors coincide with Weil-divisors, this question question can be equivalently stated in terms of Weil divisors and canonical classes: how $K_X$ and $f^*K_Y$ are related to each other? Is there any explicitly formula known?

If we think of étale maps as algebro geometric pendants to topological coverings, then in case $X$ and $Y$ smooth ($\simeq$ manifolds in topological sense) one could expect/hope that it might hold $\omega_X= f^*\omega_Y$, because in topological setting the the canonical bundles are given locally as determinant bundles of holomorphic $n$-forms and coverings are local isomorphisms.

If that's not the case in full generality for the assumptions of $X,Y$ and $f$ as above, is there at least an explicit formula à la Hurwitz for smooth curves/Riemann surfaces known relating $\omega_X$ and $ f^*\omega_Y$ to each other?

If yes, how general this formula is? Does it only hold for smooth $Y,X$? Depend it on characteristic of base field $k$?

I'm pretty sure that if we drop the étaleness assumption and so allow some even rather tame ramifications then it is nearly hopeless to expect the existence of such formula relating $\omega_X$ and $ f^*\omega_Y$ in such generality, so I hoped that maybe the étaleness assumption might provide the right amount of price which we are ready to pay to have such formula.