Many inverse problems require that solutions of differential equations satisfy appropriate properties of linear independence, for instance the independence of several solution gradients in the vicinity of a given point. In imaging applications, such solutions are often controlled from the boundary. For elliptic problems, the most useful theoretical tool to verify that such properties hold is based on an application of a unique continuation principle (UCP). For (phase space) transport equations, the control problem is quite different. We will consider transport problems with feasible or unfeasible boundary controls and will contrast these results with UCP.