  We investigate the asymptotic speed of spread of the solutions of a
non-autonomous Fisher-KPP equation with nonlocal diffusion, driven by a
thin-tailed kernel. In this paper, we are concerned with both compactly
supported and exponentially decaying initial data. For general time
heterogeneity, we provide lower and upper estimates of the spreading speed of
the solutions, which is expressed in term of the least mean of the time varying
coefficients of the problem. Under some stronger time averaging assumptions for
these coefficients, we prove that these solutions propagate with some
determined speed. In this analysis, an important difficulty comes from the lake
of regularization for the solutions arising with nonlocal diffusion. Through
delicate analysis we derive some regularity estimates (of uniform continuity
type for the large time) for some solutions of the logistic equation equipped
with suitable initial data. These results are then used to handle more general
nonlinearities and derive a rather general spreading speed results.
