Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}_l $-sheaf $(l \neq p)$ on $U$. Let $K'/K$ be any field extension and $x'$ a geometric point of $C'= C \times_K K'$ that maps to $x \in C$. Is it true that $Sw_{x'}(u'_!\mathfrak{F}') = Sw_x(u_!\mathfrak{F})$?
Here $Sw_x(u_!\mathfrak{F})$ denotes the Swan conductor of $u_!\mathfrak{F}$ at $x$ and the primes indicate the canonical pullbacks.
If $K'/K$ is algebraic, then by (Kindler, Rülling- Intro to l-adic sheaves and ramification theory, Lemma 10.2) this holds in fact true -but it is not obvious. The Swan conductor depends on representation étale fundamental groups which do not behave very well under base change. Hence I would not really expect this to hold true.
My problem is: in the paper "Semi-continuité du conducteur de Swan" Laumon seems to use the result above several times for non algebraic basechange $K'/K$.