We introduce the Distributed-order fRActional Graph Operating Network (DRAGON), a novel continuous Graph Neural Network (GNN) framework that incorporates distributed-order fractional calculus. Unlike traditional continuous GNNs that utilize integer-order or single fractional-order differential equations, DRAGON uses a learnable probability distribution over a range of real numbers for the derivative orders. This method allows for a flexible superposition of multiple derivative orders, capturing complex graph feature updating dynamics beyond the reach of conventional models.We provide a comprehensive interpretation of our flexible framework through the lens of a non-Markovian graph random walk, demonstrating its capability to capture intricate dynamics, especially when node feature dynamics are driven by a diffusion process. Furthermore, to highlight the versatility of the DRAGON framework, we conduct empirical evaluations across a range of graph learning tasks. The results consistently demonstrate superior performance when compared to traditional continuous GNN models.