Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.

Let $\Delta_X$ be the closed subscheme of $X\times X$ that is the scheme-theoretic image of the diagonal morphism $X\to X\times X$, and $\Delta_Y$ the scheme-theoretic image of the diagonal morphism $Y\to Y\times Y$.

Do we have $f^{-1}\Delta_Y=\Delta_X$ scheme-theoretically? (i.e. $\Delta_Y\times_{Y\times Y}(X\times X)=\Delta_X$)

I'd expect $\Delta_X$ to be one of the irreducible components of $f^{-1}\Delta_Y$, but was wondering about an explicit example and whether one could say something about the discrepancy between $f^{-1}\Delta_Y$ and $\Delta_X$.