  We study the long-time asymptotic behaviour of semigroups generated by
non-local Schr\"odinger operators of the form $H = -L+V$; the free operator $L$
is the generator of a symmetric L\'evy process in $\mathbb R^d$, $d > 1$ (with
non-degenerate jump measure) and $V$ is a sufficiently regular confining
potential. We establish sharp two-sided estimates of the corresponding heat
kernels for large times and identify a new general regularity property, which
we call progressive intrinsic ultracontractivity, to describe the large-time
evolution of the corresponding Schr\"odinger semigroup. We discuss various
examples and applications of these estimates, for instance we characterize the
heat trace and heat content. Our examples cover a wide range of processes and
we have to assume only mild restrictions on the growth, resp.\ decay, of the
potential and the jump intensity of the free process. Our approach is based on
a combination of probabilistic and analytic methods; our examples include
fractional and quasi-relativistic Schr\"odinger operators.
