Active transport driven by molecular motors is a key technology for the continued miniaturization of lab-on-a-chip devices, because it is expected to enable nanofluidic devices with channel diameters of less than 1 mu m and total channel lengths on the order of 1 mm. An important metric for a transport mechanism employed in an analytic device is dispersion, because it critically affects the sensitivity and resolution. Here, we investigate the mechanisms responsible for the dispersion of a swarm of "molecular shuttles", consisting of functionalized microtubules propelled by surface-adhered kinesin motor proteins. Using a simple model and measurements of the path persistence length, motional diffusion coefficient, and the distribution of average velocities, we found that, at the time scale relevant in the envisioned nanobiodevices, variations in the time-averaged velocities between shuttles will make a stronger contribution to the dispersion of the swarm than both the fluctuations around the time-averaged velocity of an individual shuttle and the fluctuations in path length due to wiggling within the channel. Overall, the dispersion of such molecular shuttles is comparable to the dispersion of a sample plug transported by electroosmotic flow.