We use quasi-two-dimensional emulsions as experimental models to study the flow of jammed materials. Our emulsions are oil droplets in water and are compressed between two parallel glass plates so that the droplets are deformed into pancake-like disks. We use microscopy to observe these droplets as they flow. From the deformed outlines of the droplets, we can measure all of the inter-droplet forces to within 10%. In this way, we study the relationship between the local stresses in the system and the rearrangements as the sample is sheared. The simplest rearrangement involves four droplets (a 'T1 event') and we confirm theoretical predictions for the quadrupolar spatial pattern of the stress redistribution around the T1 events. We also study gravity-driven flow in hoppers and investigate the probability of clogging as a function of the hopper exit size. Here, experiments and simulations show that the softness of the particles is important, as soft particles form less stable arches and thus reduce the probability of clogging.