  We study the notion of a causal time-evolution of a conserved nonlocal
physical quantity in a globally hyperbolic spacetime $\mathcal{M}$. The role of
the `global time' is played by a chosen Cauchy temporal function $\mathcal{T}$,
whereas the instantaneous configurations of the nonlocal quantity are modeled
by probability measures $\mu_t$ supported on the corresponding time slices
$\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can be
expressed in three equivalent ways: (i) via the causal precedence relation
$\preceq$ extended to probability measures, (ii) with the help of a probability
measure $\sigma$ on the space of future-directed continuous causal curves
endowed with the compact-open topology and (iii) through a causal vector field
$X$ of $L^2_{\textrm{loc}}$-regularity, with which the map $t \mapsto \mu_t$
satisfies the continuity equation in the distributional sense. In the course of
the proof we find that the compact-open topology is sensitive to the
differential properties of the causal curves, being equal to a topology induced
from a suitable $H^1_{\textrm{loc}}$-Sobolev space. This enables us to
construct $X$ as a vector field in a sense `tangent' to $\sigma$. In addition,
we discuss the general covariance of descriptions (i)-(iii), unraveling the
geometrical, observer-independent notions behind them.
