  We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for
convolution semigroups of nonlocal pseudo-differential operators. We give
general and sharp sufficient conditions under which the limits $$
  \lim_{r \to \infty} \frac{p_t(r\theta-y)}{t \, \nu(r\theta)}, \quad t \in T,
\ \ \theta \in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively
computed. Here $\nu$ is the corresponding L\'evy density, $T \subset
(0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in
$\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply
to a wide class of convolution semigroups, including those corresponding to
highly asymmetric (finite and infinite) L\'evy measures. Key examples include
fairly general families of stable, tempered stable, jump-diffusion and compound
Poisson semigroups. A main emphasis is put on the semigroups with L\'evy
measures that are exponentially localized at infinity, for which our
assumptions and results are strongly related to the existence of the
multidimensional exponential moments. Here a key example is the evolution
semigroup corresponding to the so-called quasi-relativistic Hamiltonian
$\sqrt{-\Delta+m^2} - m$, $m>0$. As a byproduct, we also obtain sharp two-sided
estimates of the kernels $p_t$ in generalized cones, away from the origin.
