A proper scheme with infinite-dimensional fppf cohomology

In algebraic geometry, very often one encounters theorems of the following flavor:

Theorem: Let \(f : X \to S\) be a proper morphism of spaces. Then for every sheaf \(\mathcal{F}\) on \(X\) that is finite, so is its pushforward \(Rf_\ast \mathcal{F}\).

Notice how I was being deliberately vague in the theorem above. What are \(X\) and \(Y\)? What does …