In developmental biology, an important problem is understanding the mechanisms underlying the formation of morphogen concentration gradients. The most commonly hypothesized mechanism involves the diffusion and degradation of morphogens from a localized source. Recently, however, an alternative mechanism has been proposed, which is based on cell-to-cell contacts mediated by thin, actin-rich cellular extensions known as cytonemes. In this paper, we develop a one-dimensional advection-diffusion transport model of cytoneme-based morphogenesis. In particular, we compare two distinct types of contact between a cytoneme tip and a target cell: direct contact and indirect contact mediated by a synapse. First, we calculate the steady-state concentration profiles and show that synaptic contacts generate broader concentration profiles, thus allowing for longer-range interactions. We then consider two alternative methods for determining how quickly the system approaches steady-state: either calculating the accumulation time using Laplace transforms, or analyzing the discrete spectrum of the associated evolution operator. The latter is a nontrivial eigenvalue problem due to the nature of the boundary conditions. Finally, we extend the direct contact model to the case of a stochastically switching boundary at the cytoneme tip, in order to take into account the fact that cytonemes dynamically grow and shrink, resulting in more temporary contacts with target cells.