  We prove a functional transcendence theorem for the integrals of algebraic
forms in families of algebraic varieties. This allows us to prove a geometric
version of Andr\'e's generalization of the Grothendieck period conjecture,
which we state using the formalism of Nori motives.
  More precisely, we prove a version of the Ax--Schanuel conjecture for the
comparison between the flat and algebraic coordinates of an arbitrary
admissible graded polarizable variation of integral mixed Hodge structures.
This can be seen as a generalization of the recent Ax--Schanuel theorems of
\cite{chiu,GaoKlingler} for mixed period maps.
