Let $f:X \to Y$ be a finite type morphism of noetherian schemes and $F$ a constructible etale sheaf on $X$. Deligne shows in SGA 4 1/2 that there exists a dense open subset $U \subseteq Y$ such that $f_*F|_U$ is constructible and is compatible with any base change.
I've tried to show that a similar property also holds for $l$-adic sheaves.So, I considered $F=(F_n)_{n \geq 1}$ an $l$-adic sheaf on $X$ (by which I mean a projective system $(F_n)_{n \geq 1}$ where each $F_i$is an $\dfrac{\mathbb{Z}}{l^i}$ sheaf such that $\dfrac{F_{n+1}}{l^nF_{n+1}} \cong F_n$
I would like to find an open subset $U$ such that for every $n \geq 1$, $f_*F_n$ is constructible, $\dfrac{f_*F_{n+1}}{l^nf_*F_{n+1}} \cong f_*F_n$ and we have compatibility with any base change for every $f_*F_n$.
I tried to use that generic base change is true for every $F_n$, but that didn't bring me up much ,as for every $f_*F_n$ the suitable open subset could change.