The notion of a flow in a network is a well studied object in combinatorics with connections to perfect matchings and max-flow/min-cut problems. We examine flows in the context of Borel dynamics, in particular in actions of translations on the torus. In this context, Borel flows have found an application in a constructive solution to Tarski's circle squaring problem. In this talk, we'll outline a proof of the existence a whole family of Borel flows in graphs generated by randomly chosen translations of the torus using ideas from discrepancy theory.