This paper considers the dynamic economic dispatch problem for a group of distributed energy resources (DERs) with storage that communicate over a weight-balanced strongly connected digraph. The objective is to collectively meet a certain load profile over a finite time horizon while minimizing the aggregate cost. At each time slot, each DER decides on the amount of generated power, the amount sent to/drawn from the storage unit, and the amount injected into the grid to satisfy the load. Additional constraints include bounds on the amount of generated power, ramp constraints on the difference in generation across successive time slots, and bounds on the amount of power in storage. We synthesize a provably correct distributed algorithm that solves the resulting finite-horizon optimization problem starting from any initial condition. Our design consists of two interconnected systems, one estimating the mismatch between the injection and the total load at each time slot, and another using this estimate to reduce the mismatch and optimize the total cost of generation while meeting the constraints.