  Motivated by diffusion processes on metric graphs and ramified spaces, we
consider an abstract setting for interface problems with coupled dynamic
boundary conditions belonging to a quite general class. Beside well-posedness,
we discuss positivity, $L^\infty$-contractivity and further invariance
properties. We show that the parabolic problem with dynamic boundary conditions
enjoy these properties if and only if so does its counterpart with
time-independent boundary conditions. Furthermore, we prove continuous
dependence of the solution to the parabolic problem on the boundary conditions
in the considered class.
