Suppose $X,Y$ are smooth projective varieties over a field $k$ with $Y$ one-dimensional, connected, and normal, and $f\colon X\to Y$ a proper, flat, surjective morphism so that its fibers are connected and generic fiber is reduced (for instance, $f$ might be a Lefschetz pencil of curves with $Y=\mathbb{P}^1$). Note that $f$ isn't necessarily smooth. Apologies for the hodgepodge of adjectives; maybe most of these assumptions aren't actually needed.
What can one say about $R^if_*\mu_{l,X}$ (take $l$ to be prime to the characteristic of $k$), where $\mu_{l,X}$ is the $l$th roots of unity etale sheaf on $X$? In particular, is it at least locally constant?
For $i>2$, these higher direct images necessarily vanish, so we only need to consider $i=0,1,2$. For $i=0$, we know by Stein factorization that $f_*\mathcal{O}_X=\mathcal{O}_Y$ and hence $f_*\mu_{l,X}=\mu_{l,Y}$ by considering the Kummer sequence. I'm not what happens for the $i=1,2$ cases though.
