Model reduction using proper orthogonal decomposition (POD) or coherent structures has a long tradition in the engineering community, dating back at least to the seminal work [4]. Using reduced-order surrogates especially pays off in a many query context arising in, e.g., the optimal control of dynamical systems. While the combination of model reduction and control has been studied extensively for parabolic(-like) systems, see, e.g., [1,2] and the references therein, standard approaches fail when dealing with transport-dominated systems that suffer from a large Kolmogorov $n$-width. Although modifications of POD have been suggested [3], a rigorous analysis of these methods in an optimal control context appears to be missing. In this contribution, a detailed study of solving optimal control problems by means of a shifted POD based reduction approach is presented. In particular, for a linear transport equation a ``first-reduce-then-optimize'' method is analyzed with respect to i) well-posedness of the (nonlinear) reduced-order system, ii) existence of an optimal control and iii) first-order necessary optimality conditions. Moreover, exploiting the underlying PDE and semigroup structures, respectively, an efficient numerical implementation is tested and compared to a standard POD based approach.