In this paper we consider a time-dependent Boltzmann equation for particle transport in biological tissues, which can be used for radiation dose calculation. We assume that the dose should be delivered to a target volume which moves over time. We formulate this as an optimal control problem and derive closed-loop control laws for the dose delivery problem using boundary and distributed control. Optimality conditions are derived. For the construction of the closed-loop control laws we use an inexact variant of model predictive control called instantaneous control. We compare numerical results obtained with instantaneous control to those obtained by optimal open-loop control, and present numerical simulations in one and two spatial dimensions. This work could be applied to image-guided radiation therapy, where patient motion during treatment is one of the future challenges.