  We prove that the set of `low rank' points on sufficiently large fibre powers
of families of curves are not Zariski dense. The recent work of
Dimitrov-Gao-Habegger and K\"uhne (and Yuan) imply the existence of a bound
which is exponential in the rank, and the Zilber-Pink conjecture implies a
bound which is linear in the rank. Our main result is a (slightly weaker)
linear bound for `low ranks'. We also prove analogous results for isotrivial
families (with relaxed conditions on the rank) and for solutions to the
$S$-unit equation, where the bounds are now sub-exponential in the rank. Our
proof involves a notion of the Chabauty-Coleman(-Kim) method in families (or,
in some sense, for simply connected varieties). For Zariski non-density, we use
the recent work of Bl\`azquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanuel
theorems for foliations on principal bundles.
