  Geometric variations of local systems are families of variations of Hodge
structure; they typically correspond to fibrations of K\"{a}hler manifolds for
which each fibre itself is fibred by codimension one K\"{a}hler manifolds. In
this article, we introduce the formalism of geometric variations of local
systems and then specialize the theory to study families of elliptic surfaces.
We interpret a construction of twisted elliptic surface families used by
Besser-Livn\'{e} in terms of the middle convolution functor, and use explicit
methods to calculate the variations of Hodge structure underlying the universal
families of $M_N$-polarized K3 surfaces. Finally, we explain the connection
between geometric variations of local systems and geometric isomonodromic
deformations, which were originally considered by the first author in 1999.
