Many biological agents transition between different biophysical states during movement. For example, proteins inside cells bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of renewal processes and derive the effective velocity and diffusivity of cargo at large time for a class of problems. We illustrate applications of the proposed method to macroscopic models of protein localization and microscopic models of cargo movement by teams of molecular motor proteins. We also show limitations of this approach in cases where the spatial dependence on microtubules is explicitly modeled.