  We consider the accelerated propagation of solutions to equations with a
nonlocal linear dispersion on the real line and monostable nonlinearities (both
local or nonlocal, however, not degenerated at $0$), in the case when either of
the dispersion kernel or the initial condition has regularly heavy tails at
both $\pm\infty$, perhaps different. We show that, in such case, the
propagation to the right direction is fully determined by the right tails of
either the kernel or the initial condition. We describe both cases of
integrable and monotone initial conditions which may give different orders of
the acceleration. Our approach is based, in particular, on the extension of the
theory of sub-exponential distributions, which we introduced early in
arXiv:1704.05829 [math.PR].
