We study the non-autonomous ordinary differential equation x = f (t, x) in the situation when the vector field f is of limited regularity, typically belonging to a space LP (O,T; Lq (JRn)). Such equations arise naturally when switching from an Eulerian to a Lagrangian viewpoint for the solutions of partial differential equations. We discuss some measurability issues in the foundations of the theory of regular Lagrangian flow solutions. Further, we examine the sensitivity of the choice of representative vector field f on solutions of the ordinary differential equation and, in particular, we demonstrate that every vector field can be altered on a set of measure zero to introduce non-uniqueness of solutions. We develop some geometric tools to quantify the behaviour of solutions, notably a non-autonomous version of subset avoidance and the r-codimension print that encodes the dimension of a subset S c JRn x [0, T] while distinguishing between the spatial and temporal detail of S. We relate this notion of dimension to the more familiar box-counting dimensions, for which we prove some new inequalities. Finally, motivated by the issues with measurability that can arise with irregular vector fields we prove some fundamental results in the theory of Bochner integration in order to be able to manipulate the representatives of the equivalence classes in LP (O,T; Lq (JRn)).