  We study the behavior of test ideals and F-singularities in families. In
particular, we obtain generic (and non-generic) restriction theorems for test
ideals and non-F-pure ideals. Additionally, we study the global behavior of
certain canonical linear systems (induced by Frobenius) associated to adjoint
line bundles, in families. As a consequence, we obtain some positivity results
for pushforwards of some adjoint line bundles and for certain subsheaves of
these.
