  We study $F$-signature under proper birational morphisms $\pi : Y \to X$,
showing that $F$-signature strictly increases for small morphisms or if $ K_Y
\geq \pi ^*K_X$. In certain cases, we can even show that the $F$-signature of
$Y$ is at least twice as that of $X$. We also provide examples of $F$-signature
dropping and Hilbert-Kunz multiplicity increasing under birational maps without
these hypotheses.
