On the Non-Medical Transport Screen with Today's Transportation created, select the route who's pickup/dropoff times you'd like to edit.
From there, you can select either "Client Pick Up" or "Client Drop Off" drop-downs. These will allow you to make adjustments to who is to be picked up/dropped off for the selected route.
When you are finished making the adjustments to the lists, be sure to select the green "Submit" button to save your changes.