  We establish a reduction principle to derive Li-Yau inequalities for
non-local diffusion problems in a very general framework, which covers both the
discrete and continuous setting. Our approach is not based on
curvature-dimension inequalities but on heat kernel representations of the
solutions and consists in reducing the problem to the heat kernel. As an
important application we solve a long-standing open problem by obtaining a
Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat
equation of the form $(-\Delta)^{\beta/2}(\log u)\leq C/t$, where $\beta\in
(0,2)$. We also illustrate our general result with an example in the discrete
setting by proving a sharp Li-Yau inequality for diffusion on a complete graph.
