Existing machine learning literature lacks graph-based domain adaptation techniques capable of handling large distribution shifts, primarily due to the difficulty in simulating a coherent evolutionary path from source to target graph. To meet this challenge, we present a graph gradual domain adaptation (GGDA) framework, which constructs a compact domain sequence that minimizes information loss during adaptation. Our approach starts with an efficient generation of knowledge-preserving intermediate graphs over the Fused Gromov-Wasserstein (FGW) metric. A GGDA domain sequence is then constructed upon this bridging data pool through a novel vertex-based progression, which involves selecting "close" vertices and performing adaptive domain advancement to enhance inter-domain transferability. Theoretically, our framework provides implementable upper and lower bounds for the intractable inter-domain Wasserstein distance, $W_p(μ_t,μ_{t+1})$, enabling its flexible adjustment for optimal domain formation. Extensive experiments across diverse transfer scenarios demonstrate the superior performance of our GGDA framework.