Morphogen protein gradients play an important role in the spatial regulation of patterning during embryonic development. The most commonly accepted mechanism for gradient formation is diffusion from a source combined with degradation. Recently, there has been growing interest in an alternative mechanism, which is based on the direct delivery of morphogens along thin, actin-rich cellular extensions known as cytonemes. In this paper, we develop a bidirectional motor transport model for the flux of morphogens along cytonemes, linking a source cell to a one-dimensional array of target cells. By solving the steady-state transport equations, we show how a morphogen gradient can be established, and explore how the mean velocity of the motors affects properties of the morphogen gradient such as accumulation time and robustness. In particular, our analysis suggests that in order to achieve robustness with respect to changes in the rate of synthesis of morphogen, the mean velocity has to be negative, that is, retrograde flow or treadmilling dominates. Thus the potential targeting precision of cytonemes comes at an energy cost. We then study the effects of non-uniformly allocating morphogens to the various cytonemes projecting from a source cell. This competition for resources provides a potential regulatory control mechanism not available in diffusion-based models.