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We investigate trajectories of microscale evaporating droplets in a stagnation point flow near a wall of a respiratory airway. The configuration is motivated by the problem of advection and deposition of microscale droplets of respiratory fluids in human airways during transmission of infectious diseases, such as tuberculosis and COVID-19. Laminar boundary layer equations are solved to describe the airflow while the equations of motion of the droplet include contributions from gravity, aerodynamic drag, and Saffman force. Evaporation is accounted for at both the droplet surfaceand the wall of the respiratory airway and is shown to delay droplet deposition as compared to the predictions of isothermal models. Evaporation at the airway wall has a stronger effect on droplet trajectories than evaporation at the droplet surface, leading to droplets being advected away by the flow and thus avoiding deposition in the stagnation point flow region.