On-lattice position jump systems, where objects move between a series of boxes, are an established tool in biological modelling. They are popular in developmental biology where they have been used to model the movement of particles and cells in the growing organism, but have also been applied to a variety of areas including cancer and cell biology. Exact stochastic simulation of such systems is computationally intensive however, and there exists a sizeable literature of methods to speed it up, whether by optimising algorithms, or developing hybrid systems incorporating elements of approximate simulation or deterministic models. In this presentation, we contribute to this work by introducing a framework for non-local diffusion on a lattice, so that objects are not limited to moving only to neighbouring boxes but can jump any distance up to a defined limit with different jump rates. We discuss how to choose these jump rates so as to produce the same results as would be obtained from a locally jumping model, but requiring significantly less computational time because fewer jumps need to be simulated. We consider when this approach is appropriate, and deal with a range of technical/implementation issues resulting from our accelerated approach, such as matching to PDEs in the continuum limit. We present simulation results throughout to validate our analysis.