  Integrable systems possess stable families of quasiparticles, which are
composite objects (bound states) of elementary excitations. Motivated by recent
quantum computer experiments, we investigate bound-state transport in the
spin-$1/2$ anisotropic Heisenberg chain ($XXZ$ chain). Specifically, we
consider the sudden vacuum expansion of a finite region $A$ prepared in a
non-equilibrium state. In the hydrodynamic regime, if interactions are strong
enough, bound states remain confined in the initial region. Bound-state
confinement persists until the density of unbound excitations remains finite in
the bulk of $A$. Since region $A$ is finite, at asymptotically long times bound
states are "liberated" after the "evaporation" of all the unbound excitations.
Fingerprints of confinement are visible in the space-time profiles of local
spin-projection operators. To be specific, here we focus on the expansion of
the $p$-N\'eel states, which are obtained by repetition of a unit cell with $p$
up spins followed by $p$ down spins. Upon increasing $p$, the bound-state
content is enhanced. In the limit $p\to\infty$ one obtains the domain-wall
initial state. We show that for $p<4$, only bound states with $n>p$ are
confined at large chain anisotropy. For $p\gtrsim 4$, also bound states with
$n=p$ are confined, consistent with the absence of transport in the limit
$p\to\infty$. The scenario of bound-state confinement leads to a hierarchy of
timescales at which bound states of different sizes are liberated, which is
also reflected in the dynamics of the von Neumann entropy.
