symplectic mapping tori. The symplectic mapping torus of an automorphism $f$ is a symplectic fibration over $T$ which has trivial monodromy in one direction, and monodromy $f$ in the other direction. Let’s say for concreteness that $T = {\mathbb{R}}^2/{\mathbb{Z}}^2$ has coordinates $(p,q)$, and that the monodromy is trivial in $q$-direction, and $f$ in $p$-direction. The symplectic mapping torus contains plenty of Lagrangian submanifolds fibered over trivial circles $\{p\} \times S^1$. If one then moves such a Lagrangian submanifold by the time-one map of the symplectic vector field $\partial_p$, the effect is the same as applying $f$ fibrewise. For suitable examples of $f$, this allows one to show that $[dq]$ does not lie in the flux group, which distinguishes the mapping torus from the trivial one (a similar approach was used in [@seidel02b]). To make a more abstract version of the argument, we consider families of objects in the Fukaya category whose deformation is driven by the class $[dq]$. The idea of introducing families of Lagrangian submanifolds into Floer cohomology theory is due to Fukaya [@fukaya02c; @fukaya09]. Generally speaking, it shows much potential for giving fundamental insights as well as applications, but also encounters considerable foundational difficulties. Here, we bypass these 