Min-cut in graph with demands/lower bounds
Max flow and min cut are dual problems, in a technical sense: there are linear programming formulations of both, and these linear programs are dual to each other. This is explained in the Wikipedia article on the max-flow min-cut theorem.

Given a variant of the max flow problem, you can find the corresponding min cut problem by writing the linear programming formulation of the max flow problem, and dualizing it. I'll let you figure out what you get in your cases.