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Consider an interacting particle system indexed by the vertices of a (possibly random) locally finite graph whose vertices and edges are equipped with marks representing parameters of the model such as the environment and initial conditions. Each particle takes values in a countable state space and evolves according to a pure jump process whose jump intensities depend on only the states (or histories) and marks of itself and particles and edges in its neighborhood. Under mild conditions, it is shown that if the sequence of (marked) interaction graphs converges locally in probability to a limit (marked) graph that satisfies a certain finite dissociability property, then the corresponding sequence of empirical measures of the particle trajectories converges weakly to the law of the marginal dynamics at the root vertex of the limit graph. The proof of this limit relies on several results of independent interest. First, such interacting particle systems are shown to be well-posed on almost surely finitely dissociable graphs, which include graphs of maximal bounded degree and any Galton-Watson tree whose offspring distribution has a finite first moment. A counterexample is provided to show that well-posedness can fail for dynamics on graphs outside this class. Second, the dynamics on a locally convergent sequence of graphs are shown to converge in the local weak sense to the dynamics on the limit graph when the latter is finitely dissociable. Finally, the dynamics are also shown to exhibit an (annealed) asymptotic correlation decay property. These results complement recent work that establishes hydrodynamic limits of locally interacting probabilistic cellular automata and diffusions on sparse random graphs. However, the analysis of jump processes requires very different techniques, including percolation arguments and notions such as (consistent) spatial localization and causal chains.