We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent on–off sources, where the on periods and off periods may have heavy-tailed probability distributions. Variants of these FCLTs hold for cumulative busy-time and idle-time processes associated with standard queueing models. The heavy-tailed on-period and off-period distributions can cause the limit process to have discontinuous sample paths (e.g., to be a non-Brownian stable process or more general Lévy process) even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M1 topology on the function space D of right-continuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M1 topology imply both heavy-traffic and non-heavy-traffic FCLTs for buffer-content processes in stochastic fluid networks.