We consider a class of long-range exclusion processes evolving either on Z or on a finite lattice with open boundaries. For the power-law jump kernel, e.g., p(x,y)=|y-x|^{-1-gamma}1_{y>x}, it is known that if p has infinite first-order moment (0<gamma<1), the particle density evolves, at superballistic space-time scale, according to a nonlocal convection equation. We extend the result to general jump kernel p=p(x,y) that is not necessarily translation-invariant. Assuming some tail bounds of p, we show that the macroscopic evolution is governed by an integro-differential equation. Some new results on the well-posedness of the regular solution will also be discussed. Based on a joint work with Patrícia Gonçalves (IST Lisbon) and Julian Kern (Freie University Berlin).