Microfluidic tools are providing many new insights into the chemical, physical and physicochemical responses of cells. Both suspension-level and single-cell measurements have been studied. We review our studies of these kinds of problems for red blood cells with particular focus on the shapes of individual cells in confined geometries, the development and use of a 'differential manometer' for evaluating the mechanical response of individual cells or other objects flowing in confined geometries, and the cross-streamline drift of cells that pass through a constriction. In particular, we show how fluid mechanical effects on suspended cells can be studied systematically in small devices, and how these features can be exploited to develop methods for characterizing physicochemical responses and possibly for the diagnosis of cellular-scale changes to environmental factors.