Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least positive, and that $\delta \subseteq |H^0(X, L)|$ is a linear system.

It is well known that just because $\delta$ is base-point-free, it does not mean that a general member defines a smooth subscheme (it need not even be reduced, for example Frobenius pull-backs of linear systems, Remark 10.9.3 in Hartshorne's algebraic geometry).

However, let's suppose the following:

$f : X \to Y$ is a map, $M$ is a very ample line bundle on $Y$ and $L = f^* M$. Suppose further that $\delta$ is the pull-back of the complete linear system $|H^0(Y, M)|$. Are there any (separability?) conditions on the map $f$ which still guarantee that Bertini holds for $\delta$? I imagine this must be well known, but I don't know the right references.

In particular, I am looking for conditions weaker than etale (/ etale outside a finite set of points)? Say in the birational case, or the finite case?