Flow of soft matter objects through one-dimensional environments is important in industrial, biological and biomedical systems. Establishing the underlying principles of the behavior of soft matter in confinement can shed light on its performance in many man-made and biological systems. Here, we report an experimental and theoretical study of translocation of micrometer-size hydrogels (microgels) through microfluidic channels with a diameter smaller than an unperturbed microgel size. For microgels with different dimensions and mechanical properties, under a range of applied pressures, we established the universal principles of microgel entrance and passage through microchannels with different geometries, as well as the reduction in microgel volume in confinement. We also show a non-monotonic change in the flow rate of liquid through the constrained microgel, governed by its progressive confinement. The experimental results were in agreement with the theory developed for non-linear biaxial deformation of unentangled polymer gels. Our work has implications for a broad range of phenomena, including occlusion of blood vessels by thrombi and needle-assisted hydrogel injection in tissue engineering.