The large-scale applicability of the micro- and nanofluidic devices demands continuous technological advancements in the transport mechanisms, especially to promptly mix the analytes and reagents at such a small scale. To this end, thermocapillarity-induced Marangoni hydrodynamics of three-layered, immiscible fluid streams in a microchannel is analytically explored. The system is exposed to periodic and sinusoidal thermal stimuli, and a theoretical framework is presented. The diffusion of the periodic thermal stimuli across and along the fluidic interfaces creates axial surface tension gradients, which induce vortical motion of the participating fluids within the microconduit. We show that depending on the physical parameters of the three participating fluids, such vortex patterns may be fine-tuned and controlled to obtain desired transport behavior. An analytical solution for the thermal and the hydrodynamic transport phenomena is obtained by solving the momentum and energy conservation equations under the umbrella of creeping flow characteristics (very low Reynolds and thermal Marangoni numbers), and nearly undeformed fluid interfaces (negligibly small Capillary number). The approximate profiles of the deformed interfaces are also quantified theoretically to justify the assumption of flat and undeformed interfaces. The independent influence of crucial thermophysical properties, the microchannel system parameters, and features of the applied thermal stimuli are shown in detail for a fixed combination of other parameters.