Understanding the mechanisms that control tissue morphogenesis and homeostasis is a central goal not only in developmental biology but also has great relevance for our understanding of various diseases, including cancer. A model organism that is widely used to study the control of tissue morphogenesis and proportioning is the Dictyostelium discoideum. While there are mathematical models describing the role of chemotactic cell motility in the Dictyostelium assembly and morphogenesis of multicellular tissues, as well as models addressing possible mechanisms of proportion regulation, there are no models incorporating both these key aspects of development. In this paper, we introduce a 1D hyperbolic model to investigate the role of two morphogens, DIF and cAMP, on cell movement, cell sorting, cell-type differentiation and proportioning in Dictyostelium discoideum. First, we use the non-spatial version of the model to study cell-type transdifferentiation. We perform a steady-state analysis of it and show that, depending on the shape of the differentiation rate functions, multiple steady-state solutions may occur. Then we incorporate spatial dynamics into the model, and investigate the transdifferentiation and spatial positioning of cells inside the newly formed structures, following the removal of prestalk or prespore regions of a Dictyostelium slug. We show that in isolated prespore fragments, a tipped mound-like aggregate can be formed after a transdifferentiation from prespore to prestalk cells and following the sorting of prestalk cells to the centre of the aggregate. For isolated prestalk fragments, we show the formation of a slug-like structure containing the usual anterior-posterior pattern of prestalk and prespore cells.