<p>Nonlocal evolution equations reproduce Turing patterns and traveling waves seen in reaction–diffusion systems. We study such equations with arbitrary integrable kernels on the periodic interval and on the real line, proving well-posedness and uniform bounds. We then develop a generalized reaction–diffusion approximation (GRDA): the solution of a nonlocal evolution equation is approximated, in Sobolev norm, by the first component of a system (GRDS) that may include advection and/or spatial-shift terms. We prove an approximation theorem: every integrable kernel can be approximated, to prescribed accuracy, by finite linear combinations of fundamental elliptic solutions; this implies the main GRDA theorem. Linear dispersion analysis and numerics further demonstrate that breaking the kernel’s even symmetry changes the onset from a Turing bifurcation (standing wave trains) to a convective Turing bifurcation producing traveling wave trains, a transition captured by the GRDA. Altogether, our results clarify when and how nonlocal interactions can be represented effectively by local GRDS dynamics.</p>