This article presents a distributed null-space-based behavioral (NSB) algorithm for the formation path-following problem of vehicles moving in three dimensions. The algorithm is applied to fleets of underactuated autonomous underwater vehicles (AUVs). The algorithm combines null-space-based control with consensus methods. First, we present a continuous-time version of the algorithm and prove its stability using Lyapunov analysis. Then, we present a discrete-time event-triggered version that, compared to similar formation path-following methods, can achieve the same steady state-error performance with fewer inter-vehicle transmissions. The effectiveness of both the continuous-time and the discrete-time algorithm is verified in numerical simulations. Furthermore, the discrete-time version is tested in experiments.