We report new dynamical modes in confined soft granular flows, such as stochastic jetting and dripping, with no counterpart in continuum viscous fluids. The new modes emerge as a result of the propagation of the chaotic behavior of individual grains-here, monodisperse emulsion droplets-to the level of the entire system as the emulsion is focused into a narrow orifice by an external viscous flow. We observe avalanching dynamics and the formation of remarkably stable jets-single-file granular chains-which occasionally break, resulting in a non-Gaussian distribution of cluster sizes. We find that the sequences of droplet rearrangements that lead to the formation of such chains resemble unfolding of cancer cell clusters in narrow capillaries, overall demonstrating that microfluidic emulsion systems could serve to model various aspects of soft granular flows, including also tissue dynamics at the mesoscale.