  We study diffusion-type equations supported on structures that are randomly
varying in time. After settling the issue of well-posedness, we focus on the
asymptotic behavior of solutions: our main result gives sufficient conditions
for pathwise convergence in norm of the (random) propagator towards a
(deterministic) steady state. We apply our findings in two environments with
randomly evolving features: ensembles of difference operators on combinatorial
graphs, or else of differential operators on metric graphs.
