  Connectivity is a fundamental structural feature of a network that determines
the outcome of any dynamics that happens on top of it. However, an analytical
approach to obtain connection probabilities between nodes associated to paths
of different lengths is still missing. Here, we derive exact expressions for
random-walk connectivity probabilities across any range of numbers of steps in
a generic temporal, directed and weighted network. This allows characterizing
explicit connectivity realized by causal paths as well as implicit connectivity
related to motifs of three nodes and two links called here pitchforks. We
directly link such probabilities to the processes of tagging and sampling any
quantity exchanged across the network, hence providing a natural framework to
assess transport dynamics. Finally, we apply our theoretical framework to study
ocean transport features in the Mediterranean Sea. We find that relevant
transport structures, such as fluid barriers and corridors, can generate
contrasting and counter-intuitive connectivity patterns bringing novel insights
into how ocean currents drive seascape connectivity.
