This work revisits the problem of distributed adaptive filtering in multi-agent sensor networks. In contrast to classical approaches, the formulation relaxes the Gaussian assumption on the signal and noise to the generalized setting of α-stable distributions that do not possess second- and higher-order statistical moments. Most importantly, the considered scenario allows for different characteristic exponents throughout the network. Drawing upon ideas from correntropy-type local similarity measures and fractional-order calculus, a novel class of distributed fractional-order correntropy adaptive filters, that are robust against the jittery behavior of α-stable signals, is derived and their convergence criterion is established. The effectiveness of the proposed algorithms, as compared to existing distributed adaptive filtering techniques, is demonstrated via simulation examples.