  The null distance for Lorentzian manifolds was recently introduced by Sormani
and Vega. Under mild assumptions on the time function of the spacetime, the
null distance gives rise to an intrinsic, conformally invariant metric that
induces the manifold topology. We show when warped products of low regularity
and globally hyperbolic spacetimes endowed with the null distance are (local)
integral current spaces. This metric and integral current structure sets the
stage for investigating convergence analogous to Riemannian geometry. Our main
theorem is a general convergence result for warped product spacetimes relating
uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of
the corresponding null distances. In addition, we show that non-uniform
convergence of warping functions in general leads to distinct limiting
behavior, such as limits that disagree.
