Differential linear repetitive processes are a distinct class of two-dimensional linear systems which can be used, for example, to model industrial processes such as long-wall coal cutting operations. Also they can be used to study key properties of classes of linear iterative learning schemes. The key feautre of interest in this paper is the fact that information propragation in one of the two separate directions in such processes evolves continuously over a finite fixed duration and in the other direction it is, in effect, discrete. The paper develops discrete approximations for the dynamics of these processes and examines the effects of the approximation techniques used on two key systems-related properties. These are stability and the structure of the resulting discrete state-space models. Some ongoing work and areas for further development are also briefly noted.