  We study the asymptotic behavior of the output states of sequences of quantum
channels. Under a natural assumption, we show that the output set converges to
a compact convex set, clarifying and substantially generalizing results in
[BCN13]. Random mixed unitary channels satisfy the assumption; we give a
formula for the asymptotic maximum output infinity norm and we show that the
minimum output entropy and the Holevo capacity have a simple relation for the
complementary channels. We also give non-trivial examples of sequences $\Phi_n$
such that along with any other quantum channel $\Xi$, we have convergence of
the output set of $\Phi_n$ and $\Phi_n\otimes \Xi$ simultaneously; the case
when $\Xi$ is entanglement breaking is investigated in details.
