We introduce a general framework for studying a class of randomized load balancing models in a system with a large number of servers that have generally distributed service times and use a first-come-first serve policy within each queue. Under fairly general conditions, we use an interacting measure-valued process representation to obtain hydrodynamics limits for these models, and establish a propagation of chaos result. Furthermore, we present a set of partial differential equations (PDEs) whose solution can be used to approximate the transient behavior of such systems. We prove that these PDEs have a unique solution, use a numerical scheme to solve them, and demonstrate the efficacy of these approximations using Monte Carlo simulations. We also illustrate how the PDE can be used to gain insight into network performance.